Princeton Review GMAT Manual

Manual for the GMAT*Exam version 8.0

All rights reserved. No part of this manual may be reproduced for distribution to a third party in any form or by any means, electronic or mechanical, including photocopying, recording, or any information retrieval system, without the prior consent of the publisher, The Princeton Review. This Manual is for the exclusive use of Princeton Review course students and is not legal for resale. GMAT is a registered trademark of the Graduate Management Admission Council. The Princeton Review is not affiliated with Princeton University or the Graduate Management Admission Council. Permission to reprint this material does not constitute review or endorsement by the Educational Testing Service or the Graduate Management Admission Council of this publication as a whole or of any other sample questions or testing information it may contain. Copyright © 2003 by Princeton Review Management, L.L.C. All Rights Reserved. 800.2Review/ www.princetonreview.com

ACKNOWLEDGMENTS

Thanks to the following for their many contributions to this course manual: Tariq Ahmed, Kristen Azzara, Shon Bayer, John Bergdahl, Marie Dente, Russ Dombrow, Tricia Dublin, Dan Edmonds, Julian Fleisher, Paul Foglino, Alex Freer, John Fulmer, Joel Haber, Effie Hadjiioannou, Sarah Kruchko, Mary Juliano, Jeff Leistner, Sue Lim, Michael Lopez, Stephanie Martin, Chas Mastin, Elizabeth Miller, Colin Mysliwiec, Magda Pecsenye, Dave Ragsdale, “GMAT” Jack Schieffer, Cathryn Still, Fritz Stewart, Rob Tallia, Tim Wheeler, Stephen White, and the staff and students of The Princeton Review. Special thanks to Adam Robinson, who conceived of many of the techniques in this manual.

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TABLE OF CONTENTS

INTRODUCTION Welcome The TPR Plan Structure of the Course Make the Commitment

The GMAT How a CAT Works

PRE-CLASS ASSIGNMENTS Assignment 1 Subjects, Verbs, and Pronouns Sentences Phrases and Clauses Subject-Verb Agreement Pronouns Verb Tenses Summary Answers and Explanations Math Fundamentals Math Vocabulary Solving Equations and Inequalities Translating from English to Math Summary Drill Answers and Explanations Admissions Insight

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Assignment 2 Modifiers, Parallel Construction, and Idioms Modifiers Parallel Construction Idioms Summary Answers and Explanations Part-Whole Relationships Fractions Decimals Percents Probability Interest Data Sufficiency Ranges Summary Drill Answers and Explanations

Assignment 3 Arguments Parts of an Argument Common Flaws Summary Drill Answers and Explanations Ratios and Statistics Ratios Averages Rates Median, Mode, and Range Standard Deviation Data Sufficiency Tricks and Traps Summary Drill Answers and Explanations Admissions Insight

Assignment 4 Reading Comprehension Passage Topics Use Your Scratch Paper Work the Passage Translate the Question Paraphrase the Answer Summary Drill Answers and Explanations Exponents, Roots, and Factors Distribution

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Exponents Roots Quadratics Factorials Summary Drill Answers and Explanations Admissions Insight

Assignment 5 Inferences and Paradoxes Inference Questions Resolve/Explain Questions Summary Drill Answers and Explanations Geometry Can You Trust the Diagram? Lines and Angles Triangles Quadrilaterals Circles Coordinate Geometry Solid Geometry Summary Drill Answers and Explanations AWA Scoring The Basic Approach The Argument Task Sample Admissions Insight

Assignment 6 Sentence Correction Revisited Beyond the Big 6 Harder Verbal Problems Harder Sentence Correction Questions Answers and Explanations Math Miscellaneous Groups, Functions, Sequences, and Summation Harder Permutations and Combinations Summary Drill Answers and Explanations Analysis of an Issue Essay Basic Approach Issue Task Sample Admissions Insight

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Assignment 7 Harder Verbal Questions Harder Arguments Questions Harder Reading Comprehension Harder Math Complexity Difficult Phrasing Tricks and Traps Answers and Explanations Parting Words Mental Preparation Count Down to the GMAT At the Test Center Admissions Insight

IN-CLASS LESSONS Lesson 1 Welcome Understanding Your Score Set Your Goals Introduction to Pacing Verbal Introduction Sentence Correction 1 The Format Basic Approach The Big Six Subject/Verb Agreement Verb Tense Pronouns Red Pencil Fever Math Fundamentals Math Introduction Data Sufficiency Algebra vs. Arithmetic Plugging In Hidden Plug Ins Plugging In the Answers Homework Review Practice Answers and Explanations

Lesson 2 Sentence Correction 2 Misplaced Modifiers Parallel Construction Idioms

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Math 2 Keep Plugging Away Yes/No Data Sufficiency Percent Change Probability Test Smarts Homework Review Practice Answers and Explanations

Lesson 3 Critical Reasoning 1 Basic Approach Assumption Questions Weaken Questions Strengthen Questions Identify-the-Reasoning Questions Math 3 Ratios Averages Rates Standard Deviation Pieces of the Puzzle Simultaneous Equations Test Smarts Pacing: The Beginning Test 2 Goals Test Analysis Homework Review Practice Answers and Explanations

Lesson 4 Reading Comprehension The Basic Approach Question Types Specific Questions General Questions Pace Yourself Wisely Math 4 More Roots and Exponents More Factors and Factorials More Quadratics More Yes/No Data Sufficiency Permutations and Combinations Test Smarts Homework Review Practice Answers and Explanations

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Lesson 5 Critical Reasoning 2 Inference Questions Resolve/Explain Questions Minor Question Types Math 5 Right Triangles Coordinate Geometry Solid Geometry Drill Analysis of an Argument The Template Test Smarts Pacing: The Middle Test 3 Goals Test Analysis Homework Review Practice Answers and Explanations

Lesson 6 Sentence Correction Revisited Beyond the Big Six Harder Sentence Correction Questions Sentence Correction Review Math 6 Functions Sequences Groups Harder Probability Questions Harder Permutations and Combinations Analysis of an Issue The Template Test Smarts Pacing: The End Test 4 Goals Test Analysis Homework Review Practice Answers and Explanations

Lesson 7 Arguments and RC Revisited Critical Reasoning Review Harder Arguments Questions Harder Reading Comprehension Math 7 Complex Problems Practice Answers and Explanations

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Introduction

INTRODUCTION

WELCOME THE TPR PLAN Welcome to The Princeton Review! We’re delighted that you chose us to help you prepare for the GMAT. This course provides two important benefits: • A review of the subjects tested by the GMAT • Test-taking strategies to help you maximize your performance The Math section of the GMAT tests concepts from high school arithmetic, algebra, and geometry. The Verbal section measures your reading, grammar, and logical reasoning skills. The Analytical Writing Assessment essays test your organization and writing skills. You have seen most of these topics previously, in high school or elsewhere. However, you probably haven’t used these skills in quite some time. This course will review these areas and introduce you to a few topics you may not have seen before. Use the multiple-choice format to your advantage.

In addition to reviewing the material tested by the GMAT, you will learn testtaking strategies to help you best use your knowledge within the format of the test. You will learn the traps the test writers set for you and learn how to avoid them. You will learn how to use the multiple-choice format to your advantage. We’ll cover everything you need to know to meet the specific challenges posed by the GMAT.

STRUCTURE OF THE COURSE The Princeton Review GMAT course consists of six parts: pre-class assignments, class time, homework, diagnostic tests, the online student center (OSC), and help sessions. You’ll get the most from the course if you take full advantage of each component.

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GMAT MANUAL

Pre-class Assignments Prior to each class, your teacher will assign a pre-class section from the manual. These sections review core concepts and skills that your teacher will build on in class. Be sure to complete your reading assignment before every class.

Classes Bring your materials to every class.

We will show you how to approach every type of question in every section and how to construct a pacing plan. Each week you will learn new skills and concepts and have a chance to review the homework and material from the previous class. Be sure to bring your materials (the course manual, the Official Guide and any other handouts) to every class. Your GMAT class will have no more than eight students. Take advantage of the intimate setting by asking questions, getting involved in the class discussions, and letting your instructor know about your particular needs.

Homework In addition to your pre-class assignment, your instructor will assign homework each week. You’ll work practice problems from the manual and the Official Guide and you’ll also do online practice each week. Follow the guidelines below to get the most from your homework. Practice using the methods you learn in class.

Use the techniques you learn in class. The techniques we teach work, but some may feel awkward at first. It is extremely important to get comfortable using the techniques and methods on questions as opposed to merely understanding the concepts. Keep up with the homework. Your teacher will expose you to ideas in a logical order, and you will miss out if you fall behind. Develop an awareness of your timing. Be conscious of time from the very beginning. Make the small but important investment in a digital timer that can both count down and count up. Your local Radio Shack is a good source for timers. Set it to count up from zero when you begin work on a set of problems. Note how long it takes you to complete a set of ten questions. The purpose will not be to hit a particular target, but rather to make you conscious of how long it takes you to do different types of problems. As you progress through the course, knowing your capabilities will make setting your ultimate pacing strategy that much easier. Resist the temptation to check the answer after each question. Instead, complete at least ten questions before you check answers. Develop your ability to concentrate. The GMAT requires intense concentration for extended periods; use your homework sessions to develop this ability. On the actual exam, you’ll need to be able to work for 75 minutes at a time without a break. Try scheduling your practice time in blocks of 75 minutes. If you develop the habit of working steadily for the entire period, you’ll have the stamina needed for the exam.

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Introduction

Track your progress. Simply doing problems will not result in significant score improvements. You need to learn from the problems you do as you go along, whether or not you get them right on the first try. Review your work to determine why you got questions right or wrong, and look for patterns in your performance. Based on what you observe, adjust your strategy on the next set of questions. Make a list of the problems you want to discuss and bring that list to class. Your teacher can help you out, but he or she is not a mind reader. Leave your calculator at the door. The first thing to get used to about the Math portion of the GMAT is that calculators are not permitted. Most of us depend on calculators for basic mathematical computations such as balancing a checkbook. A sure way to increase your math skills is to get accustomed to life without a calculator. Work out everything on scratch paper. Soon, working math without a calculator will seem like second nature.

Knock the rust off your math skills!

Practice Tests Practice tests are an extremely important component of your GMAT course. In addition to your first practice exam, you will take several Computer Adaptive GMATs. Your instructor will tell you when to take these tests. These practice tests serve a couple of important functions. First, they give you the opportunity to become familiar with the structure and format of the test. Pacing is essential, and you need a chance to develop a sense of timing. Taking a standardized test is a skill just like any other; it requires practice. Second, these tests allow you and your instructor to monitor your progress and target areas that need improvement. Please be aware that ETS has not disclosed its exact scoring method. Scores from our practice tests (and all other simulated GMATs) should be viewed as approximate predictions of your GMAT score. After you take a test, spend some time reviewing your performance. Look at the questions you missed and use the explanations to help you understand the correct answer. Print copies of your score report and any questions you want to discuss and bring them to class. Your instructor will either address them in class or set up an extra-help session to go over them. You will receive a users’ guide that explains the details of using the tester program. If you have any technical problems with the practice tests, please call the technical support phone number in the users’ guide or contact us at [email protected].

Online Student Center (OSC) The Princeton Review’s Online Student Center (OSC) has much more than online practice tests. The Your Course section contains online lessons aligned with the classroom lessons. Use these lessons to review and reinforce what you learn in class. The Practice section has targeted drills that allow you to practice working specific types of problems. The more problems you work on the computer, the more comfortable you’ll feel at your exam. The Guide to the Online Student Center explains how to access all the OSC features.

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Extra Help If after doing the pre-class assignments, attending all classes, and doing the homework, you are still having trouble with some portion of the GMAT, speak to your instructor. He or she can set up a help session outside of class time.

MAKE THE COMMITMENT Bottom Line: We will show you proven techniques for cracking the GMAT. It is up to you to learn and practice them until you are comfortable using them under the pressure of taking an exam. We’d love to wave a magic wand and raise your GMAT score. Unfortunately, it doesn’t work that way. Improving your GMAT score requires a lot of hard work. Your teacher will be there to support and guide you, but it’s up to you to make preparing for the GMAT a priority. This class involves a substantial amount of work outside the scheduled class sessions. Plan to spend seven to ten hours per week outside of class on pre-class assignments and homework. You’ll need to schedule additional time during weeks in which practice tests are assigned. Reserve practice time in your schedule. If you schedule several blocks of time throughout the week, you’ll get the work done and make the most progress. If you tell yourself, “I’ll work on it when I have some free time,” you are likely to fall behind quickly. We’ve worked with tens of thousands of GMAT students over the years, and we know that students who practice regularly get the best results.

THE GMAT Business schools use the GMAT (Graduate Management Admission Test) to predict the performance of students applying for MBA programs. The admissions staff will consider your GMAT score, undergraduate GPA, work experience, recommendation letters, and application essays in making admissions decisions.

Who Writes the GMAT? As you may already know, ETS—the same folks who ruined your high school years with PSATs, SATs, and Achievement Tests (SAT IIs)—is responsible for the GMAT. The folks at ETS write most of the other exams for graduate study, including the GRE (for graduate schools), as well as exams for CIA agents, barbers, golf pros, and travel agents. ETS is a private, nonprofit corporation (though it does have highly profitable for-profit divisions). It is not supervised by the government. It is not supervised by anyone, at any level. What gives ETS the right to administer this test? The fact that it gives this test.

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Introduction

Structure of the GMAT The GMAT consists of three sections: an essay section (Analytical Writing Assessment or AWA), a multiple-choice Math section, and a multiple-choice Verbal section. Your test session will look something like this:

Section

Time

# Questions

Type of Questions

AWA Essay #1

30 Minutes

1 Topic

Essay (typed)

AWA Essay #2

30 Minutes

1 Topic

Essay (typed)

5-minute break

Math

75 Minutes

37 Questions

Multiple-choice

5-minute break

Verbal

75 Minutes

41 Questions

Multiple-choice

On each GMAT exam, you’ll be given two essays — Analysis of an Issue (the issue essay) and Analysis of an Argument (the argument essay). Each of the AWA essays contains one question, and you will have 30 minutes to answer it by typing an essay into the computer. The word processor is rudimentary, with only cut, paste, and delete functions. Typing speed is really not much of a factor because your essay will only be three to five paragraphs in length. However, if you do not type at all, you should spend some time getting comfortable with using a keyboard. The Math section contains two types of questions: problem solving and data sufficiency. Problem solving questions are the typical multiple-choice math questions that you know from the SAT and other standardized tests. The data sufficiency questions are less familiar. They test the same topics (arithmetic, algebra, and geometry), but the format is different. You will learn about this format and how to approach it systematically in the first class. You can expect that 50 to 60 percent of the math questions will be problem solving, and 40 to 50 percent will be data sufficiency. The Verbal section includes three types of questions: sentence correction, critical reasoning, and reading comprehension. Sentence correction questions test your ability to spot grammatical mistakes. Critical reasoning questions test your ability to understand and analyze arguments. Reading comprehension questions test your ability to find information in a long passage.

Experimental Questions The Math section includes nine experimental questions, and the Verbal section includes eleven experimental questions. These questions do not count toward your score. Why do you have to answer questions that don’t affect your score?

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The test writers need to try out new questions to ensure that they are wellwritten and produce the desired results. In other words, you pay for the privilege of serving as a research subject every time you take a GMAT. There is no way to identify which questions are experimental. They are sprinkled randomly throughout each section. Don’t waste time during the test guessing whether a question is experimental, but keep in mind that approximately one-fourth of the questions are unscored.

Scratch Paper Avoid mistakes. Write out your work on scratch paper.

Because this test is presented on a computer screen, you will not be able to write on the problem to label diagrams, scratch off answers, circle key words, and so forth. Instead, you are required to do all of your work on the scratch paper provided at the test center. Do not try to work out problems in your head! That is a sure-fire way to make careless mistakes. You will be provided with six sheets of scratch paper, stapled together, when you begin the test. While it is possible to request more, you may lose time in getting the proctor’s attention in order to exchange your old scratch paper for new paper. Instead, try to fit all of your work for at least one section on the sheets provided. During the break between sections, you can exchange your used sheets for a fresh set if necessary. Start practicing using your scratch paper now. When you work homework problems in this manual, do not circle words or write notes directly on the problem. Instead, write everything off to the side, as if you were using separate scratch paper. Get a notebook to use when you work problems from the Official Guide or do online drills and tests. Set up each page just as you would your scratch paper during an exam, and do all your work in the notebook. Be sure to label the problems so that you can review them easily once you have finished.

Scores The questions and presentation of your GMAT are carefully designed to produce results which, when analyzed, produce a bell curve. In other words, very few people get a perfect score, and equally few people get every question wrong. The majority of us wind up in the middle, somewhere between 200 and 800. Since ETS attains these results each time it administers the test, it claims that the GMAT is an accurate gauge of our abilities. As you’ll soon see, that’s pretty far from the truth. Your GMAT score actually consists of several different numbers, each of which covers a part of your performance on the GMAT. The most familiar number is the overall, or composite, score. This is the number you have seen in all the business school rankings and similar literature. It ranges from 200 to 800 in 10point increments. Your composite score is determined from a combination of your scores on the Math and Verbal sections of the test. A score of 540 ranks in the 50th percentile, meaning half of all examinees score above that level and half score below that level. GMAT scores are valid for up to five years. You also receive separate Verbal and Math subscores, which theoretically range from 0 to 60 for each section. In practice, scores below 10 or above 50 are rare. The mean (average) Verbal score is 27, and the mean Math score is 35. These are scaled scores, which means that if two people each score a 34, they show

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Introduction

comparable ability, even if they had totally different questions on their respective exams. Most schools are not very interested in your Verbal and Math subscores unless there is a significant gap between the two. Your Analytic Writing Assessment (AWA) essays are graded on a scale of 0–6. There are two essays, and two readers (one human, one computer) look at each one. ETS then averages all four scores and rounds the result to the nearest half point. For example, if you get a 4 and a 5 on your argument essay and two 5’s on your issue essay, your final AWA score is 5.0. The average AWA score is 4.0, and about 75 percent of all students score between 3 and 5. This score does not factor into your composite score.

The AWA score is not very important for most applicants. Focus on the more important areas.

Every score—overall, Verbal, Math, and AWA—is also accompanied by a percentile, so you can determine how well you compare with other test takers. An overall score with a percentile of 76 means that 76 percent of all people who have taken this test in the past four years or so did worse than you did. It also means that 24 percent did better. Your results from the test will look something like this: Math

%

Verbal

%

Overall

%

AWA

%

35

48

30

59

550

54

4.5

62

HOW A CAT WORKS How does the computer determine your score? The GMAT is a CAT, or computer adaptive test. The operative word is adaptive. The level of difficulty of the test questions adapts to match your performance. In other words, when you answer a question correctly, the next question will be harder. When you answer a question incorrectly, the next question will be easier. A CAT looks at several things to calculate your score for a section: • Number of questions you answer correctly • Difficulty of the questions you answer • Number of questions you complete When a section starts, the computer doesn’t know anything about you, so it estimates that you have medium ability. You start with a medium score, and the first question is of medium difficulty. Every time you answer a question correctly, the computer raises your score and gives you a more difficult question. Every time you answer a question incorrectly, the computer lowers your score and gives you an easier question. The computer recalculates your score after every question. The difficulty level of the next question generally matches your current score. However, the computer also has to meet certain requirements for the types of questions in a section. For example, the Math section has to have a balance of problem solving and data sufficiency questions and also has to have the proper mix of arithmetic, algebra, and geometry. Thus, the difficulty of a particular question may not exactly match your current score because these factors also affect the choice of question. In general, though, the question difficulty tracks your performance.

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60

Score/Difficulty

50 40 Amy

30

Brian 20 10 0

0

1

2

3

4

Look at the example in the graph above. Amy and Brian both take a fourquestion CAT section. Amy starts off well. She answers the first question correctly and her score increases. The difficulty of the questions is also increasing at this point. She gets the second question right, and her score and level of difficulty, increase again. The third question is even more difficult, and Amy gets the question wrong. What happens here is very important. Her next question will be easier than the last one, but not easier than the first one, because she has already “earned” a certain level of difficulty. For the same reason, her final score will drop from where it was, but will not drop from where she started. Amy gets the fourth question wrong. Her score will again decrease, as will her level of difficulty, but it will not drop below that of the first question. Brian, however, starts by getting questions wrong. He answers the first question incorrectly, thereby lowering his score and decreasing the level of difficulty of the next question. He gets the second question wrong and continues to decrease the level of difficulty and his final score. The third question that Brian sees is easier than the questions he saw before, and he gets the question right. Again, what happens at this point is very important. His next question will be harder than the last one, but not harder than the first one. His score will increase from where he was, but not from where he started. Brian gets the fourth question right. His score will again increase, as will his level of difficulty, but it will not surpass that of the first question. Amy and Brian each answer two questions correctly and two questions incorrectly. However, who ends up with the higher score? Amy does. Why? Except for the first question, each of the questions she answered was harder than those that Brian answered. Even though their performances might lead you to believe that their scores would end up the same, Amy will finish with a much higher score. Earlier questions count more than later questions do.

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Because the level of difficulty of the questions that you answer is a factor in computing your final score, it is important that you achieve the highest level of difficulty you can early in the test. In effect, you must do well at the beginning of the test.

Introduction

No Skipping Allowed On a paper-and-pencil test, you can skip a problem and return to it later. On a CAT, however, that’s not possible. The CAT requires you to answer each question before moving on to the next one. Also, you cannot go back to a question once you have answered it.

Pacing Knowing how a CAT works is the first step in developing a pacing plan. If you divide the time that you have to take the test by the number of questions on the exam, you will find that you have about 1.8 minutes per verbal question and 2 minutes per math question. However, because all questions are not created equal you should not spend the same amount of time on each question. Proper pacing is essential to success on the GMAT. This course will cover many ways to help you pace the test in order to maximize your performance. Learning how to pace yourself is just as important as learning all of the material. The most important guidelines to remember are these: 1. Start slowly and carefully. Eliminate careless mistakes. 2. Gradually pick up speed so that you can finish the section. 3. Don’t waste time on killer questions. Guess and move on. As demonstrated in the example with Amy and Brian, how you perform at the beginning of the section greatly affects your final score. Your score can fluctuate dramatically depending on how many questions you get right or wrong. By the end of each section, however, the computer has already determined the possible range for your final score. Your score will fluctuate only within a narrow range. What does this mean in practical terms? The earlier questions are the most important, so slow down and do your best on them. While the early questions carry the most weight, that doesn’t mean you should focus solely on them. If it were the case that you could work the first half of the questions, guess on the second half, and get a great score, everybody would take that approach. To separate examinees into scoring levels, the GMAT confronts you with more questions than most people can comfortably complete in the time allowed. It rewards those who not only complete the early questions correctly, but also those who complete a greater number correctly. Expect to feel pressed for time when you take a test. Know that this is normal, and stay calm. The test penalizes you if you do not answer every question in a section. Remember that the total number of questions answered affects your score. If you leave a question unanswered, you did not complete it, nor did you get it right. Therefore, the penalty for leaving any question blank is quite severe. Finish the section, even if it means that you have to guess randomly at the end of the section. Granted, if you randomly answer questions, your score will not be as high as that of someone who correctly answers those questions, but it will still be higher than if you left those same questions blank. You can (and many do) score in the 700s and still guess on a few questions. If you answer questions correctly, the subsequent questions get increasingly harder. You are virtually guaranteed to see a few questions that you will not be able to solve. For these killer questions, don’t waste your valuable time sitting and staring. Take your best guess and move on. Spend your time on questions that you can solve.

Do not leave anything blank.

Don’t get stuck on killer questions!

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You’ll learn much more about constructing a pacing plan in class, and you’ll refine your pacing strategy as you take practice tests.

Register for the GMAT Register for the GMAT soon, since seats at the preferred times tend to fill up quickly. While individual needs vary, plan to take your exam no more than one month after your final Princeton Review class session. To register for the GMAT, call GMAC at 1-800-GMAT-NOW (1-800-462-8669) or register online at www.mba.com.

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PRE-CLASS ASSIGNMENTS

ASSIGNMENT 1

SUBJECTS, VERBS, AND PRONOUNS The test writers claim that GMAT sentence correction questions test your knowledge of grammar. They don’t, at least not in any truly substantive way. Remember your grammar handbook from middle school? It was huge. The number of rules that the GMAT could possibly test is infinite. The good news is that the GMAT tests only a limited number of grammar rules. We won’t overwhelm you with unnecessary grammar terminology or make you spend hours diagramming sentences. We’ll concentrate only on the grammar rules tested on the GMAT. In class, your instructor will discuss how the GMAT tests each rule and show you the best way to attack the questions.

The GMAT tests only a few basic grammar rules.

In order to discuss the rules of grammar, you need to be familiar with a few concepts. This section reviews the building blocks of sentences and introduces some grammar rules related to subjects, verbs, and pronouns.

SENTENCES A sentence is a group of words that expresses a grammatically complete thought. The basic parts of the sentence are the main subject and main verb. Look at an example of a very simple sentence: Julian reads. The main subject is Julian, and the main verb is reads.

Subjects The main subject is the noun that performs the main action. The subject can be a single word or a group of words. The examples below illustrate different types of subjects. Single noun

The cloak was made of velvet.

Gerund

Sleeping for eight hours a night is important to me.

Infinitive

To learn how to read music requires diligence.

Noun clause

That the dog had bitten her gave her reason to fear it.

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Verbs Verbs express action, condition, or state of being. The main verb in a sentence is the word that expresses the main action.

PARTS OF SPEECH Consider a more complex version of the first example: Because he wants to improve his GMAT score, Julian very dutifully reads the extremely important pre-class assignments in this manual. This example tells us much more about who Julian is, what he reads, how he reads, and why he reads. We’ll use it to review the parts of speech.

Nouns Nouns are people, places, and things. Subject nouns, like Julian, perform an action. Object nouns, like assignments, receive the action or are objects of prepositions (see below). Assignments is the object of the verb reads. Manual is the object of the preposition in. We will discuss the various kinds of nouns in more detail later.

Pronouns Pronouns take the place of nouns and are used to avoid repetition. Like the nouns they replace, pronouns can function as either subjects or objects. In the sentence above, he is the subject of wants.

Modifiers Modifiers describe, or modify, other words in a sentence. Adjectives describe nouns. In the sentence above, important and pre-class modify assignments. Adverbs modify verbs, adjectives, and other adverbs. They usually provide information about where, when, or how something happens. In the sentence above, dutifully modifies reads, very modifies dutifully, and extremely modifies important.

Prepositions Prepositions are the little words that show relationships between other words or phrases. They usually create prepositional phrases that act as modifiers. In the sentence above, in is a preposition, and in this manual is a prepositional phrase that modifies assignments. Manual is the object of the preposition.

Conjunctions Conjunctions connect words or parts of sentences. In the sentence above, because connects he wants to improve his GMAT score with Julian very dutifully reads the extremely important pre-class assignments in this manual.

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Assignment 1

PHRASES AND CLAUSES Phrases and clauses are groups of words that act as modifiers or nouns. Let’s examine the distinction between phrases and clauses.

Phrases A phrase is a group of words that acts as a part of speech, not a complete sentence. Modifying phrases take on the role of adjectives or adverbs. The prepositional phrase in this manual acts as a modifier in our earlier example. It describes the location of the pre-class assignments. Noun phrases can be the subject or object of a sentence, as illustrated in the examples below. Preparing for the GMAT requires hard work. Allen wanted to leave. In the first example, the phrase preparing for the GMAT is the subject. To leave is the object of the second example, and it tells us what Allen wanted.

Clauses A clause is a group of words that has a subject and a verb. Main clauses can stand alone as complete sentences. Dependent clauses cannot stand alone as complete sentences. In the example above, Julian very dutifully reads the extremely important pre-class assignments in this manual is the main clause, and Because he wants to improve his GMAT score is the dependent clause. Clauses can also act as subjects, objects, or modifiers. Conjunctions such as and, but, and or often link two independent clauses to create one sentence: I went to the movies, and she went to the library. Notice that each of the clauses expresses a complete thought and could stand alone as a complete sentence. On the other hand, a dependent clause does not express a complete thought, even though it has a subject and a verb. A dependent clause functions as a noun, an adjective, or an adverb. She is the woman who was wearing the leopard-skin coat last night. I don’t know why you bothered to come to class without your homework. Notice that these clauses cannot stand alone as sentences; they depend on independent clauses to form grammatically complete thoughts. The first dependent clause functions as an adjective modifying woman. In the second example, the clause is the object of the verb know. Dependent clauses begin with that, whether, if, because, or the words we use to begin questions (who, whom, whose, what, where, when, why, which, how).

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Quick Quiz: Subjects and Verbs In the following sentences, circle the main subjects and underline the main verbs. 1. What you see is what you get. 2. The unexamined life is not worth living. 3. Commuting by bicycle helps people enjoy the benefits of fresh air and exercise. 4. His courage as a pilot of a U2 spy plane earned Gary Powers a posthumous citation. 5. Made from a single log, a dugout canoe draws very little water. 6. Felicia and Tim went to the same high school.

SUBJECT-VERB AGREEMENT Subject-verb agreement is one grammar concept tested on the GMAT. The basic rule for subject-verb agreement is straightforward. Singular subjects take singular verbs, and plural subjects take plural verbs.

Don’t Be Fooled Deciding whether a subject is singular or plural can sometimes be challenging. Abstract Nouns Some nouns describe a quality, idea, or state of being. These abstract nouns, such as sadness, truth, laughter, poverty, and knowledge, represent a single thing. Abstract nouns are singular.

Justice always prevails. Wealth is nice, but happiness is better. Collective Nouns Collective nouns name a group of things, animals, or people. The group has individual members, but it’s a single entity.

Collective nouns are singular.

The committee votes on the budget this week. The school of fish swims around the reef. The family that just moved in next door is nice. Verb Forms as Nouns The -ing form of a verb (also known as the present participle) can be used as a noun, and in such cases it is called a gerund. When the to form of a verb acts like a noun, we call it an infinitive noun.

Gerunds and infinitive nouns are singular.

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Walking is great exercise. To err is human; to forgive is divine.

Assignment 1

Nouns That End in -s Most of the time, a noun that ends in -s is plural, but some singular nouns end in -s. If you’re unsure whether a noun is singular or plural, ask yourself if it represents one thing or several things. (Hint: country names are always singular.) That species has a number of interesting habits. Economics is one of my favorite subjects.

Not all nouns that end in -s are plural.

The Netherlands is a country in Europe.

Singular Pronouns Some singular pronouns, such as everybody or no one, are easy to mistake for plural pronouns. Even though we often treat these as plural words in everyday speech, the GMAT writers use more formal rules and define them as singular. Each of the witnesses was questioned by the police. Everyone in the senior class is sick with the flu.

Any pronoun that ends in -body, -thing, or -one is singular.

Either of the restaurants is fine with me. In the first example, the subject is Each, not witnesses. Witnesses is the object of the preposition. Similarly, Either is the subject of the third example, not restaurants. When the subject of a sentence is a pronoun followed by a prepositional phrase, the pronoun is the main subject. Make sure the pronoun, not the object of the preposition, agrees with the verb. Either and neither are singular when they serve as the subject of a sentence.

These pronouns are singular and take singular verbs: no one

nobody

nothing

someone

somebody

something

everyone

everybody

everything

anyone

anybody

anything

none

each

Compound Subjects When a subject includes more than one noun, we call it a compound subject. My best friend and her sister are very similar in personality. When and joins two subjects, you must use a plural verb. However, compound subjects joined by or, either. . .or, and neither. . .nor follow a different rule. In these cases, the verb agrees with the noun closest to it. Neither the bride nor the groom was able to remember the names of all the guests. Neither Joe nor his cousins were happy on the first day of school.

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The Number and A Number When you refer to a number of things, you’re talking about many things, and you need a plural verb. When you refer to the number of things, you’re talking about one particular number, and you need a singular verb. The number of bad movies showing this summer is unbelievable. A number of my friends are going to the beach this weekend.

Quick Quiz: Subject-Verb Agreement Circle the appropriate verb in the parentheses below. 1. Gloria and Calvin (are, is) no longer friends. 2. The number of times I have told you I do not want to go to the concert with you (amaze, amazes) me. 3. Samantha, in addition to Carrie, Charlotte, and Miranda, (is, are) going to the beach on Saturday. 4. Neither Mark nor his neighbors (is, are) able to open the doors to the patio. 5. Next month, Jack and Chrissy, along with Janet, (is, are) moving to the larger apartment upstairs. 6. Tom’s family (is, are) considering whether there (are, is) any affordable places to go on vacation in Europe. 7. Each of the boys (is, are) overwhelmed by the amount of work to be done. 8. Every one of the golf balls (has, have) been hit into the sand trap. 9. This sandwich is the only one of all the sandwiches made at the deli that (is, are) inedible. 10. Skiing is an example of a sport that (is, are) best learned as a child.

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Assignment 1

PRONOUNS Pronouns are words used in place of nouns, and they are usually used to avoid repetition. In the previous sentence, the word they replaces the word pronouns. Some pronouns function as subjects, while others function as objects. Another set of pronouns indicates possession. Subject Pronouns

Object Pronouns

Possessive Pronouns

I

me

my

you

you

your

he/she/it

him/her/it

his/hers/its

we

us

our

they

them

their

who

whom

whose

Pronoun Agreement Read the following paragraph and underline all the pronouns: This weekend, Matt is throwing a party to celebrate his birthday. He has invited many friends and family. Matt’s sister, Teresa, is bringing the cake. She has promised him that it will be chocolate. Each pronoun agrees with the noun it replaces. He, his, and him all refer to Matt. She refers to Teresa. It replaces cake. Just as subjects and verbs must agree, pronouns must agree in number with the nouns they replace. The noun a pronoun replaces is called the antecedent. Use a singular pronoun to replace a singular noun, and use a plural pronoun to replace a plural noun. The same types of nouns that make subject-verb agreement tricky can cause problems for pronoun agreement. Look at the following examples: Everyone should do their homework. The golden retriever is one of the smartest breeds of dogs, but they cannot do your GMAT homework for you. In both examples, the use of their or they is incorrect. Since everyone is singular, it must be paired with a singular pronoun, such as his or her. Either his or her would be considered correct in this case since we have no in formation about gender. In the second example, they refers to the golden retriever. Since the antecedent retriever is singular, the pronoun it should be used in the second clause.

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Pronoun Ambiguity In addition to agreeing with the noun it replaces, a pronoun must clearly refer to only one noun. If the pronoun could conceivably refer to more than one noun, you face the problem of pronoun ambiguity. Pronouns must unambiguously refer to a single noun. Look at the following example: Lisa Marie was supposed to meet Jen at the museum at eleven, but she was late. It is unclear to whom she refers, Lisa Marie or Jen. The sentence should be rewritten to clear up the confusion. If Jen were the one who arrived late, the corrected sentence would read: Lisa Marie was supposed to meet Jen at the museum at eleven, but Jen was late.

Pronoun Consistency Pronoun usage must also be consistent. Look at the following example: One should watch your purse on a crowded subway. The sentence seems to be saying that some unknown person should be on the lookout for your purse on a crowded subway. For the purposes of the GMAT, sentences that refer to an undefined person can use either you or one, as long as a single pronoun is used consistently. The example above could be rewritten two different ways: One should watch one’s purse on a crowded subway. You should watch your purse on a crowded subway.

Quick Quiz: Pronoun Agreement Correct the pronoun agreement in the following sentences. Not all sentences have an error. 1. Each of the chefs makes their own special dish. 2. I still keep my diary and scrapbooks from childhood because they remind me of my youth. 3. A student must see their advisor before turning in his thesis. 4. The person who stole my bicycle is a thief. 5. One should always look where he is going when you cross the street. 6. In 1980, the Netherlands agreed to limit fishing in certain Atlantic Ocean beds, but in 1981, they terminated the agreement. 7. The flock of seagulls flew overhead before it swooped down and settled on the water.

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Assignment 1

Who vs. Whom These two words cause lots of trouble, but they shouldn’t. If you’re confronted with a choice between who and whom, look at the role played by the pronoun in the sentence. If the pronoun is performing an action, it’s the subject, and you should use who. If the pronoun is acted upon by the verb or follows a preposition, it’s an object, and you should use whom. The detective knows who committed the murder. To whom should I speak about the matter? If you find yourself struggling with the choice between who and whom, try replacing the pronoun with she or her. If a sentence should use she, use who. If a sentence should use her, choose whom. For example, it would be correct to say, “The detective knows she committed the murder,” because she is a subject pronoun. Similarly, it would be correct to say, “I should speak to her about the matter.” The first example needs a subject pronoun, and the second example needs an object pronoun.

VERB TENSES Verb tense places an action in time, and the basic tenses are past, present, and future. The examples below illustrate the tenses you’ll encounter in GMAT sentences. Present

I study, I am studying, I have studied

Past

I studied, I had studied, I was studying

Future

I will study, I will be studying, I will have studied

The variations within the basic categories of past, present, and future allow us to express ideas more precisely. If a tense uses a helper verb, such as a form of to be or to have, use the helper verb to determine the tense. For example, I was walking is in the past tense because was denotes the past. I am walking is in the present because am denotes the present. Let’s look at the variations in more detail. Knowing the names of the tenses isn’t necessary, but you need to be able to classify them as past, present, or future.

Present The simple present expresses a habitual action, a fact, or something that is happening now. Beth runs three miles every morning. Both baseball games are on television right now. When you want to describe something that’s in progress right now, use the present progressive tense. (It’s sometimes called the present continuous tense.) Present progressive uses a form of “to be” followed by the -ing form of the verb (also known as the present participle). The kids on the playground are laughing loudly.

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The present perfect tense describes an action that started at an indefinite time in the past and either continues into the present or has just been completed. This tense uses the helping verb has or have followed by the past participle. Astrid has read a book a week since she was twelve. I have never been to Spain.

Past The simple past indicates a completed action or condition. I wrote my final paper over the weekend. Use past progressive to describe an action that was ongoing in the past. We were sleeping when the fire alarm went off. Use the past perfect when you want to make it clear that one action in the past happened before another. This tense requires the helping verb had. Before she began college last fall, she had never been more than twenty miles from home. In the example above, the past perfect action was cut off by an intervening event in the more recent past. The past perfect cannot stand alone as the only verb in a sentence.

Future Simple future, as you would expect, describes an action that will take place in the future. This tense requires the use of the helper verb will. I will clean my room tomorrow. Use future progressive to describe an ongoing action that takes place in the future. The tense is formed by using the future form of the helping verb to be plus the –ing form of a verb. I will be cleaning my room when you arrive. Use future perfect to indicate an action that will be completed by a specified time in the future. We will not have finished dinner by the time you arrive.

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Assignment 1

Quick Quiz: Verb Tense Choose the correct verb tense in the parentheses for each sentence below. 1. Yesterday afternoon, clouds rolled in, the sky grew ominous, and thunder (was, is) heard in the distance. 2. Before the union leadership even began salary negotiations, it (had made, made) up its mind to stand firm in its position. 3. The Boy Scouts (love, loved) their new clubhouse, which they built last summer. 4. My new co-workers (had been, were) very friendly to me until they learned my salary was considerably higher than theirs. 5. Roberts already (finished, had finished) the experiments by the time Fuller made the discovery in his own laboratory. 6. Since 1980, several economies in developed nations (are experiencing, have experienced) declines and recoveries. 7. The belief in vampires (was first recorded, had first been recorded) in the early fifteenth century. 8. By the time Spanish explorers first encountered them, the Aztecs (have developed, had developed) the calendar. 9. Unlike the brown sparrow, the passenger pigeon (was slaughtered indiscriminately, had been slaughtered indiscriminately) and became extinct in 1914.

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SUMMARY Subjects and Verbs • Singular subjects take singular verbs, and plural subjects take plural verbs. • Abstract nouns, collective nouns, and verb forms acting as nouns are singular. • Pronouns that end in -body, -one, or -thing are singular. • The number is singular. A number is plural.

Pronouns • Pronouns must agree in number with the nouns they replace. • Pronouns must unambiguously refer to only one noun. • Who is a subject pronoun. Whom is an object pronoun.

Tense • The basic tenses are past, present, and future. • Sentences should stay in one tense unless the action takes place at two different times.

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Assignment 1

ANSWERS AND EXPLANATIONS Quick Quiz: Subjects and Verbs The subjects are in italicized text, and the verbs are underlined. 1. What you see is what you get. 2. The unexamined life is not worth living. 3. Commuting by bicycle helps people enjoy the benefits of fresh air and exercise. 4. His courage as a pilot of a U2 spy plane earned Gary Powers a posthumous citation. 5. Made from a single log, a dugout canoe draws very little water. 6. Felicia and Tim went to the same high school.

Quick Quiz: Subject-Verb Agreement The subjects are in italicized text, and the verbs are underlined. 1. Gloria and Calvin are no longer friends. 2. The number of times I have told you I do not want to go to the concert with you amazes me. 3. Samantha, in addition to Carrie, Charlotte, and Miranda, is going to the beach on Saturday. 4. Neither Mark nor his neighbors are able to open the doors to the patio. 5. Next month, Jack and Chrissy, along with Janet, are moving to the larger apartment upstairs. 6. Tom’s family is considering whether there are any affordable places to go on vacation in Europe. 7. Each of the boys is overwhelmed by the amount of work to be done. 8. Every one of the golf balls has been hit into the sand trap. 9. This sandwich is the only one of all the sandwiches made at the deli that is inedible. 10. Skiing is an example of a sport that is best learned as a child.

Quick Quiz: Pronoun Agreement If a correction was needed, the original pronoun has been crossed out and replaced with the correct pronoun. 1. Each of the chefs makes their his own special dish. The pronoun here refers back to the noun each. You could also use her. 2. I still keep my diary and scrapbooks from childhood because they remind me of my youth. No error. They replaces both diary and scrapbooks. 3. A student must see their his advisor before turning in his thesis. The student must be male, because it is his thesis, therefore it must also be his advisor.

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4. The person who stole my bicycle is a thief. No error. Who is the subject of the clause who stole my bicycle and is used correctly. 5. One should always look where he one is going when you one crosses the street. Be consistent. 6. In 1980, the Netherlands agreed to limit fishing in certain Atlantic Ocean beds, but in 1981, they it terminated the agreement. The Netherlands is a single country. 7. The flock of seagulls flew overhead before it swooped down and settled on the water. No error. It agrees with the noun flock.

Quick Quiz: Verb Tense The correct verb is underlined. 1. Yesterday afternoon clouds rolled in, the sky grew ominous, and thunder was heard in the distance. There is no reason to switch verb tense, and all the other verbs (rolled, grew) are in the past tense. 2. Before the union leadership even began salary negotiations, it had made up its mind to stand firm in its position. Past perfect is the correct tense here because, while both actions occurred in the past, one action (had made) occurred before the other. 3. The Boy Scouts love their new clubhouse, which they built themselves last summer. Presumably they still love their clubhouse, so it’s okay to switch from the past tense to the present tense. 4. My new co-workers had been friendly to me until they learned my salary was considerably higher than theirs. Past perfect is the best tense here because both events happened in the past, but one happened before the other. 5. Roberts already had finished the experiments by the time Fuller made the discovery in his own laboratory. Past perfect is the best tense here because both events happened in the past, but one happened before the other. 6. Since 1980, several economies in developed nations have experienced declines and recoveries. Present perfect is the best tense because the declines began in the past and continue into the present. 7. The belief in vampires was first recorded in the early fifteenth century. The simple past is best here because the sentence describes an action in the past that has been completed. 8. By the time Spanish explorers first encountered them, the Aztecs had developed the calendar. Past perfect is the best tense here because both events happened in the past, but one happened before the other. 9. Unlike the brown sparrow, the passenger pigeon was slaughtered indiscriminately and became extinct in 1914. The simple past is best here because the sentence describes an action in the past that has been completed.

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Assignment 1

MATH FUNDAMENTALS The math portion of the GMAT tests nothing but “basic” math skills, which sounds easy enough until you realize that this is exactly how ETS hopes to get you—that is, by testing you on terms and concepts that you haven’t dealt with since high school.

MATH VOCABULARY In order to beat ETS at its own game, you need to make sure that you understand several core concepts and terms. Many questions on the GMAT are unanswerable unless you know what these terms mean.

Do you know these terms? Fill in as much of the chart as you can before turning the page to check the answers.

Term

Definition

Examples

Integer Positive Negative Even Odd Sum Difference Product Divisor Dividend Quotient Prime Consecutive Digits Distinct Absolute Value

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Definition

Term

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Examples

Integer

A whole number that does not contain decimals, fractions, or radicals. Integers can be negative, positive, or 0.

–500, 0, 1, 28

Positive

Greater than 0

Negative

Less than 0

Even

An integer that is divisible by 2

–40, 0, 2

Odd

An integer that is not divisible by 2

–41, 1, 3

Sum

The result of addition

The sum of 3 and 4 is 7.

Difference

The result of subtraction

The difference of 7 and 2 is 5.

Product

The result of multiplication

The product of 2 and 7 is 14.

Divisor

The number you are dividing by

8 ÷ 2 = 4 (2 is the divisor.)

Dividend

The number you are dividing into

8 ÷ 2 = 4 (8 is the dividend.).

Quotient

The result of division

8 ÷ 2 = 4 (4 is the quotient.)

Prime

A number that is divisible only by itself and 1. 2, 3, 5, 7, 11 Negative numbers, 0, and 1 are NOT prime.

Consecutive

In order, not necessarily ascending

–1, 0, 1 or 10, 9, 8

Digits

0–9; the numbers on the phone pad

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Distinct

Different

2 and 3 are distinct; 4 and 4 are not distinct

Absolute Value

The distance from 0 on a number line. The absolute value is always positive. the symbol “ ” means absolute value.

5 3

0.5, 25, –72.3,

−7 , –2 4

4 = 4 ; −4 = 4

Assignment 1

Use the definitions to solve the problem below. 1. If x and y are distinct negative integers greater than –10, what is the greatest possible product of x and y? 2 4 72 81 90 First you have to figure out which numbers fit the definitions for distinct, negative, integer, and greater than –10. The only numbers that work are –9, –8, –7, –6, –5, –4, –3, –2, and –1. The question asks for the product, the result of multiplying x and y. A negative times a negative is positive, so any two of these numbers multiplied together will be positive. That means that the greatest possible value of xy is –9 × –8, or 72. The answer is (C).

Quick Quiz: Positive/Negative and Even/Odd Do you know the rules of positive/negative and even/odd? (Answers to all questions are found at the end of each chapter.) Circle one: 1. Negative × or ÷ negative = positive/negative 2. Positive × or ÷ positive = positive/negative 3. Negative × or ÷ positive = positive/negative 4. Even × even = even/odd 5. Odd × odd = even/odd 6. Even × odd = even/odd 7. Even + or – even = even/odd 8. Odd + or – odd = even/odd 9. Even + or – odd = even/odd

Factors and Multiples A factor is a positive integer that divides evenly into another positive integer. The factors of 12 are 1 and 12, 2 and 6, and 3 and 4. You can also think of factors as the numbers you multiply together to get a product. A multiple is the product of some positive integer and any other positive integer. For example, the multiples of 12 are 12, 24, 36, 48, 60. . . The largest factor of any number, and the smallest multiple of any number, is always the number itself.

A number has few factors, but many multiples.

Let’s try another question: 2. What is the sum of the distinct, prime factors of 60? 9 10 11 12 30

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A prime factor is a factor that is also a prime number. The factor tree is a great tool for figuring out the prime factorization of a number.

60 2

30 2

15 3

5

Thus, 60 = 2 • 2 • 3 • 5, the distinct prime factors of 60 are 2, 3, and 5, and (B) is the answer.

Rules of Divisibility Sometimes it’s helpful to know whether one number is divisible by (i.e., a factor or divisor of) another. Learning these rules will save you precious time on the test.

A number is divisible by

Rule

Center

2

It's even (i.e., its last digit is even)

1,576

3

Its digits add up to a multiple of 3

8,532

8 + 5 + 3 + 2 = 18

4

Its last two digits are divisible by 4

121,532

32 ÷ 4 = 8

5

Its last digit is 5 or 0

568,745

320

6

Apply the rules of 2 and 3

55,740 It's even and 5 + 5 + 7 + 4 + 0 = 21

9

Its digits add up to a multiple of 9

235,692 2 + 3 + 5 + 6 + 9 + 2 = 27

10

Its last digit is zero

11,130

12

Apply the rules of 3 and 4

3,552 3 + 5 + 5 + 2 = 15 and 52 ÷ 4 = 13

Let’s try a problem. 3. If x is an integer divisible by 15 but not divisible by 20, then x CANNOT be divisible by which of the following? 6 10 12 30 150 If x is divisible by (i.e., a multiple of) 15 but not 20, then x could be, for instance, 15, 30, or 45, but not 60. If a number is divisible by 15, then it is also divisible by factors of 15, i.e. 3 and 5. Just looking at the rule of divisibility for 3, (A), (B), and (D) are out. Let’s look at answer choice (E): 150 is divisible by 15 but not 20; thus x could be 150 and (E) is out. So by POE, the answer must be (C).

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Assignment 1

Quick Quiz: Math Vocabulary Fill in the ovals for all the terms that must be true. Integer

Even Odd Positive

Negative Prime

–1 0 1 2 3 0.5

3 0.001 51 Even Positive × Odd Negative = Even Negative × Even Negative = Odd Negative × Even Positive = Odd Positive × Odd Positive =

Order of Operations Sometimes you’re given a long, ugly arithmetic equation to solve, and you have to know the correct order of operations. Does “Please Excuse My Dear Aunt Sally” ring a bell? What exactly does PEMDAS stand for anyway? P|E|M D|A S P stands for parentheses. Solve expressions in parentheses first. E stands for exponents. Solve expressions with exponents next. M stands for multiplication, and D stands for division. Do all the multiplication and division together in the same step, going from left to right. A stands for addition, and S stands for subtraction. Do all the addition and subtraction together in the same step, going from left to right.

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Quick Quiz: Order of Operations Solve the following: 1. 17 – 11 × 3 + 9 = 2. (17 – 11) × (3 + 9) = 3. –5(3 – 7)2 + 94 = 4. 54 ÷ 6 × 3 – 3(5 – 13) = 5. 54 ÷ (6 × 3) – 3(5 – 13)2 =

SOLVING EQUATIONS AND INEQUALTIES You’ll learn a lot about how to avoid most algebra in the coming weeks, but you should still know a little about variables and equations. A variable is a letter, such as x, that represents an unknown amount.

Equations with One Variable The simplest equations have no exponents and are called linear equations. Let’s try an example. If 3x+

x

+ 7=21, then x =

2 To solve this equation, we need to get the variable by itself on one side of the equation. Since the fraction is awkward, let’s begin by multiplying each term on both sides of the equation by 2: x

2(3 x +

2

+ 7) = (212 )

That gives us: 6x+ x +14 = 42 When we combine the x’s, we get: 7x +14 = 42 To isolate the variable, we need to subtract 14 from each side: 7x +14 –14 = 42 – 14 That leaves: 7x = 28 Finally, we can divide both sides by 7 to solve for x: 7x 7

=

28 7

Now we know that x = 4. Do the same thing to both sides of an equation.

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To solve any linear equation with one variable, isolate the variable on one side of the equation and the numbers on the other side. The process may require multiplication, division, subtraction, or addition of different numbers.

Assignment 1

Try this one on your own before you read the explanation.

1. If

5 3

=

a 2

and b =

7

, then a+2b=

3 10 3 14 3 8 9 14

In this question we’re asked to find the value of a + 2b. The question gives us the value of b. However, we have to solve the first equation in order to find the value of a. Solve by cross multiplying. Simply multiply opposing numerators and denominators, and set them equal to one another: 3×a=5 × 2 3a = 10 a=

10 3

Now, we have enough information to answer the question. All we have to do is perform a substitution. Simply substitute the values of a and b into a + 2b. 10 7 24 Since + 2( ) = = 8 , the answer is (C). 3 3 3

Inequalities Let’s review symbols that describe inequalities:

≠

means is not equal to

>

means is greater than

<

means is less than

≥

means is greater than or equal to

≤

means is less than or equal to

Solve single-variable inequalities in exactly the same way as you do singlevariable linear equations, with one additional rule: If you multiply or divide an inequality by a negative number, you must flip the inequality sign.

Flip the inequality sign when you multiply or divide by a negative number.

Let’s work an example: 25 < –7x + 4 ≤ 60 To find the possible values for x, we need to isolate the variable. Begin by subtracting 4 from each part of the inequality: 25 – 4 < –7x + 4 – 4 ≤ 60 – 4 That gives us: 21 < –7x ≤ 56 © Princeton Review Management, L. L. C.

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To isolate x, divide each part of the inequality by –7: 21 −7

<

−7 x −7

≤

56 −7

Because we’re dividing by a negative number, we need to flip the inequality signs, which leaves us with: –3 > x ≥ –8 The solution tells us that x is greater than or equal to –8 and less than –3.

Simultaneous Equations Some GMAT problems involve sets of equations with more than one variable. These are known as simultaneous equations. To solve multiple equations, add or subtract the equations so that one of the variables (preferably the one you don’t need) cancels out, leaving you with a one-variable equation. Let’s look at an example: If 2x – 3y = 14 and x + 3y = 4, what is the value of y? Begin by stacking the equations. If we add the equations, the quantity 3y drops out of the sum. 2x – 3y = 14 x + 3y = 4 3x = 18 x=6 Now plug x = 6 into either equation to get y = –

2 . 3

You must have at least as many distinct linear equations as you have variables in order to solve for all the variables. The rule above does not by itself tell you how to solve for the variables, but it does tell you whether you have enough information to solve for each of the variables, which is very useful for working data sufficiency questions. Consider the equations below: x + 7y = 24 3x + 3y = 18 Do you have enough information to solve for x and y? Yes. We have two equations with the same two variables. Consider another set of equations: z + 7y = 24 3x + 3y = 18 Do you have enough information to solve for x and y? In this case, we don’t have enough information to solve for all the variables because there are 3 variables but only 2 equations. Try one more set of equations: x + 3y = 24 72 – 9y = 3x

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Assignment 1

Do you have enough information to solve for x and y? If you subtract 3y from both sides of the first equation, and then multiply both sides by 3, you will get 3x = 72 – 9y, which is the equivalent of the second equation above. Thus, you don’t have two distinct equations and cannot solve for the variables.

Two equations are not distinct if one is a multiple of the other.

TRANSLATING FROM ENGLISH TO MATH The GMAT tests more than your ability to perform calculations and solve equations. You will frequently encounter word problems that test your ability to translate between English and math. It’s easy to become confused by long problems with confusing phrasing. Avoid this trap by translating word problems in bite-sized pieces. Let’s look at a sample expression: The product of x and y is five less than four times as much as one-third of z. Taken as a whole, the sentence might make your head spin. Let’s take it one piece at a time. A product is the result of multiplication, so the product of x and y means xy. Is means equals, so now we know: xy = five less than four times as much as one-third of z. 1 z . Four times as much as means multiply by four. Adding 3 the new information, we get:

One-third of z means

1  xy = five less than 4  z  3  That leaves only five less than to translate. Less than tells us to subtract, so we need to subtract five from the last part of the expression. Now we have the equation:

1  xy = 4  z  –5 3  Use the table below as a guide to translating some common words and phrases.

English

Math

More than, greater than, sum of

Addition (+)

Less than, fewer than, difference between

Subtraction (–)

Times as many/much as, times more than, of, the product of

Mult iplication (×)

Goes into, divided by, quotient of

Division (÷)

Is, are, was, were equals, the same as

Equals (=)

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Quick Quiz: Translation Translate the following expressions into their mathematical equivalents. 1. 5 more than a. 2. x is 18 more than y. 3. Dan has 9 fewer pencils than Jeff does. 4. w is 8 less than p. 5. Half of x is 10. 6. The number of boys in the class is one-third the number of girls. 7. k is three times greater than l. 8. The shirt cost five times as much as the pants. 9. Twice a certain number is equal to that number minus 10. 10. A certain number divided by three is equal to 9 more than that number.

SUMMARY This chapter reviewed a lot of terms and rules. If you’re having any difficulty remembering them, make a set of math flashcards. Work with the cards until you know the rules backward and forward. In the coming weeks, you can add to your flashcards as you encounter additional math concepts.

Math Vocabulary • Learn the math terms presented in the lesson. • Know the rules of positive/negative, even/odd, divisibility, and order of operations. • Use a factor tree to find the prime factorization of a number.

Solving Equations and Inequalities • Linear equations are equations with no exponents. • Solve linear equations and inequalities by isolating the variable. Any operations performed on one side of an equation or inequality must also be performed on the other side. • If you multiply or divide by a negative number, you must flip the inequality sign. • When you work with simultaneous equations, you must have at least as many distinct equations as you have variables in order to solve for all the variables. Equations that are multiples of one another are not distinct.

Translating from English to Math • Translate word problems in bite-sized pieces.

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Assignment 1

DRILL 1. The product of two integers is 36 and their sum is 13. What is the positive difference between the two numbers? 1 4 5 7 9

6. If 2x is 12 less than the sum of 6x and 4y, then x +y= 1 2 3 6 9

2. If r is the remainder when 15 is divided by 4, what is the remainder when 17 is divided by r? 0 1 2 3 4

7. If

3. What is the sum of the distinct prime numbers between 50 and 60? 104 108 110 112 116

8. If 4r + 3s = 7, 2r + s = 1, and 2r + 2s = t – 4, what is the value of t? 6 8 10 12 It cannot be determined from the information given.

4. If

2 5

of x is 8, then what is

1

< –5, then

3 y < –4 y < 11 y > –4 y>4 y > 11

of x?

4 2 3.2 4 5 16

5.

7 − 2y

9. How many integers between 1 and 200, inclusive, are divisible by both 3 and 4? 8 12 15 16 24

a = (b – 2) x + 2 In the equation above, x is a constant. If a = 14 when b = 5, what is the value of a when b = 7? –84 –22 10 22 30

10. How many three-digit integers between 310 and 400, exclusive, are divisible by 3 when the tens digit and the hundreds digit are switched? 3 19 22 30 90

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ANSWERS AND EXPLANATIONS Quick Quiz: Positive/Negative and Even/Odd (circle one) 1.

Negative × or ÷ negative = positive/negative

2.

Positive × or ÷ positive = positive/negative

3.

Negative × or ÷ positive = positive/negative

4.

Even × even = even/odd

5.

Odd × odd = even/odd

6.

Even × odd = even/odd

7.

Even + or – even = even/odd

8.

Odd + or – odd = even/odd

9.

Even + or – odd = even/odd

Quick Quiz: Math Vocabulary Fill in the ovals for all the terms that must be true. Integer –1 0 1 2 3 0.5 3

0.001 51 Even Positive × Odd Negative = Even Negative × Even Negative = Odd Negative × Even Positive = Odd Positive × Odd Positive =

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Even Odd Positive

Negative Prime

Assignment 1

Quick Quiz: Order of Operations Solve the following: 1.

17 – 11 × 3 + 9 = –7

2.

(17 – 11) × (3 + 9) = 72

3.

–5(3 – 7)2 + 94 = 14

4.

54 ÷ 6 × 3 – 3(5 – 13) = 51

5.

54 ÷ (6 × 3) – 3(5 – 13)2 = –189

Quick Quiz: Translation 1.

a+5

2.

x = y + 18

3.

D= J −9

4.

w =p−8

5. 6.

1 x = 10 2 1 b= g 3

7.

k = 3l

8.

s = 5p

9.

2x = x − 10

10.

x =x+9 3

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GMAT MANUAL

Drill 1. C

2. C

Let’s work through this systematically. “Product” means the result of multiplication and “sum” means the result of addition, so we need to find two numbers that give you 36 when you multiply them and 13 when you add them: 1 × 36 = 36, but 1 + 36 is not 13; 2 × 18 = 36, but 2 + 18 is not 13; 3 × 12 = 36, but 3 + 12 is not 13; 4 × 9 = 36, and 4 + 9 = 13, so 4 and 9 are the numbers we’re looking for. “Difference” means the result of subtraction: 9 – 4 = 5. The remainder is what’s left over after you divide: 15 ÷ 4 = 3 remainder 3. So r = 3, and 17 ÷ 3 = 5 remainder 2, so the answer is 2.

3. D Every prime number other than 2 must be odd, so let’s list the odd numbers between 50 and 60: 51, 53, 55, 57, 59. Which of these numbers are prime? Since a prime number is a number only divisible by 1 and itself, we need to figure out whether any of these numbers are divisible by a number other than 1 and itself. Think about small numbers. Obviously, none of these numbers is divisible by 2, but are any of them divisible by 3? (Remember your divisibility rule for 3: sum of the digits is divisible by 3.) 51 is divisible by 3 because the sum of the digits is 6, and 57 is divisible by 3 because the sum of the digits is 12. Therefore, 51 and 57 are not prime. Looking at the remaining numbers, 55 is divisible by 5 because it ends in 5, so 55 is not prime either. That only leaves 53 and 59, and the sum of those numbers is 112, leaving us with (D). 2 4. D Translate and solve for x. If • x = 8, then 5 5 1 x = 8 • = 20. Thus • 20 = 5. 2 4 5. D Since x is a constant, it always represents the same value. Plug in the values given for the other variables and solve for x. Thus you get 14 = (5 – 2)x + 2, so 14 – 2 = 3x, and x = 4. Now plug that into your equation for x, and plug in your new value for b and you get a = (7 – 2)4 + 2, so a = 22.

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6. C

Let’s translate the English into math. If 2x is 12 less than the sum of 6x and 4y, that means 2x = 6x + 4y – 12. Putting all our variables on one side, we get 12 = 4x + 4y, so x + y = 3.

7. E

Multiply both sides by 3, and you get 7 − 2 y < −15 . Subtract 7 from both sides to isolate the variable ( −2 y < −22 ). Finally, divide both sides by −2, and you have y > 11. Don’t forget to flip the inequality sign when you multiply or divide by a negative number.

8. C

To isolate t, we need to get rid of r and s. Stack the equations and multiply every term in the second and third equations by −1. Then, add the equations. That gives us: 4r + 3s = 7 –2r – s = –1 –2r – 2s = –t + 4 0 = –t + 10 Thus, t must be 10.

9. D You can count out the multiples of 12 (especially if it’s early in the section and accuracy is crucial): 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, 156, 168, 180, and 192. Or, divide 200 by 12 (it goes in 16 times with a remainder of 2). 10. D This one is somewhat sadistic because that last part of the question is a red herring. Since the sum of the digits dictates whether a number is divisible by 3, the order of the digits doesn’t matter. Once you get past that, the only answer choice that comes close is 30 because there are 89 numbers between 310 and 400 and every third one must be a multiple of 3.

Assignment 1

ADMISSIONS INSIGHT The Princeton Review and PrincetonReview.com can help you with the business school admission process.

Finding the Right Schools If you don’t already have a list of potential business schools in mind, it’s not too early to start browsing. Start thinking about where you might want to apply before you finish the course so you’ll be ready to go when your scores come in. The first step in finding the best business school fit is to determine the direction you hope to take after graduation. Business school programs vary widely in their offerings and strengths. Pinpointing the exact reason for choosing a career in business and a specific field of interest will help in selecting a school. Look for schools that offer a broad-based curriculum if you haven’t decided on a particular field. Choosing a particular path also affects your candidacy; admissions committees favor applicants who have clear goals and objectives. Moreover, once at school, students who know what they want make the most of their two years. Without a targeted direction, opportunities for career development—such as networking, mentoring, student clubs, and recruiter events—are squandered. Begin researching business schools now on PrincetonReview.com. Using our powerful search tools online, the business school search can be narrowed down to those schools that best meet your priorities and interests. Our Advanced Business School Search tool provides a list of schools that match indicated needs and preferences. Use it to find a business school, compare top choice schools, or browse an alphabetical list of b-schools across the U.S. Our School Match tool is a chance to actually connect with schools. Once you complete the information section—academic record, work history, priorities in school, etc.—interested schools can contact you. Keep an open mind; a smaller or less familiar school may have just the right atmosphere, specialization, or career network. Our School Profiles produce facts and figures on admissions, academics, student body, and career outlook. And My Review lets you manage the entire application process online. Research business schools, then save the schools you wish to target in a personal list. It will also keep track of your applications and their deadlines, store results from our financial aid tools, and more.

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Keep in mind: • Too many applicants rely on the magazine rankings to decide where to apply. But statistics rarely show the whole picture. Don’t buy into any “bests.” Simply seek out the program that is the best for you. • Get a feel for the spirit of the student body at prospective schools. Even more so than undergraduates, business students rely on one another both inside and outside the classroom. Employ the same strategies used to research undergraduate colleges: Talk to students, alumni, and teachers; visit campuses; sit in on classes; and meet professors. These initial contacts are even more important for business school. Business school benefits are part educational, part networking. Establishing relationships early on may help your job search down the road. • Choosing the best MBA program involves a lot more than looking at a school’s average GMAT scores. On the one hand, just because a school has an average GMAT score that resembles yours doesn’t mean that it would be a good fit. Even the best schools have their own strengths and weaknesses. Plus, GMAT scores and other quantitative measures are only part of what admissions officers consider. If your background and goals don’t match up with those of the school, don’t count on being admitted, even with impressive numbers.

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ASSIGNMENT 2

MODIFIERS, PARALLEL CONSTRUCTION, AND IDIOMS This lesson reviews important concepts related to idioms, modifiers, and parallel construction.

MODIFIERS As you learned in the last lesson, modifiers are words that describe, or modify, other words in a sentence. Adjectives modify nouns; adverbs modify verbs, adjectives, and other adverbs. Single words or entire phrases can be modifiers.

Quick Quiz: Modifiers Underline the modifying words and phrases in the sentences below, and draw arrows to what they modify. 1. Walking down the avenue, I was caught in a torrential downpour. 2. Left in the refrigerator for several weeks, the meat was now spoiled. 3. Michael Jordan, who is now 40 years old, is still considered one of the best players in professional basketball.

Misplaced Modifiers In the movie Animal Crackers, Groucho Marx says, “I once shot an elephant in my pajamas. How he got into my pajamas I’ll never know.” The humor comes from a grammatical error (and you thought grammar was no fun). Though in my pajamas is meant to describe I, it seems to describe the elephant. A misplaced modifier is created when a modifier is not adjacent to the thing it’s intended to modify. Misplaced modifiers on the GMAT most often appear in sentences that begin with an introductory phrase. Consider the following example: Excommunicated by the Roman Church in 1521, the Protestant Reformation was led by Martin Luther.

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GMAT MANUAL

The sentence begins with the modifying phrase Excommunicated by the Roman Church in 1521. Since the noun that immediately follows the phrase is the Protestant Reformation, the sentence implies that the Reformation was excommunicated. To correct the misplaced modifier, we need to rewrite the sentence: Excommunicated by the Roman Church in 1521, Martin Luther led the Protestant Reformation. Now the sentence tells us that Martin Luther was excommunicated, and the misplaced modifier has been corrected.

Quick Quiz: Misplaced Modifiers Decide if the modifying phrase in each of the sentences below is placed correctly or if it is misplaced. 1. Arranged in secret, the discovery of Romeo and Juliet’s marriage was made only after their deaths. (no error, misplaced modifier) 2. Discovered by Marie Curie and Pierre Curie, polonium and radium were first isolated in 1898. (no error, misplaced modifier) 3. Invented by James Hargreaves in 1765, the spinning jenny was capable of spinning eight to eleven threads at one time. (no error, misplaced modifier) 4. I overheard him say that he had cheated on the exam while I was standing in the hallway. (no error, misplaced modifier) 5. Once a very powerful nation, France’s status has declined in recent years. (no error, misplaced modifier)

Quantity Words Quantity words that describe nouns raise another modifier issue. Some nouns refer to concrete things, such as children, tables, or dollars, and are countable. Other nouns refer to abstract ideas or amorphous things, such as air, beauty, or money, and are non-countable. Different quantity words apply to countable and non-countable nouns. Countable

Not Countable

fewer

less

number

amount or quantity

many

much

Here are some samples of the proper uses of these quantity words: If there were fewer cars on the road, there would be less traffic. The number of cars on the road contributes to the amount of traffic. There’s too much traffic on this road because there are too many cars. 44

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Assignment 2

Another issue involves comparisons. There are two separate situations: comparing two things and comparing three or more things. Memorize these rules:

Two Things

Three or More Things

-er

-est

more

most

between

among

Examples: Between you and me, I am taller. Among the four of us, I am the tallest.

Quick Quiz: Quantity Words Circle the correct form of the quantity word in the sentences below. 1. The (better, best) you prepare for the GMAT, the (higher, highest) your score will be. 2. (Many, most) of the population lives in poverty. 3. (Many, most) of the people live in poverty. 4. Since I withdrew money from my bank account, the (number, amount) of dollars in the account is now (fewer, less). 5. Since I withdrew money from my bank account, the (number, amount) of money in the account is now (fewer, less). 6. Some people consider the Yankees to be the (greatest, greater) baseball team ever.

PARALLEL CONSTRUCTION When a sentence includes a list or comparison, each word or phrase in the list or comparison must have the same grammatical structure. The following examples illustrate parallel construction: A melody is a succession of single tones that vary in pitch, harmony, and rhythm. By the time he was thirteen, Mozart had not only composed sonatas, but he had also performed before royalty. Walking briskly can be as aerobically beneficial as jogging. Her novel was praised as an exciting story, a social critique, and a philosophical inquiry.

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Quick Quiz: Parallelism Choose the word that creates parallel construction. 1. On Saturday, David had to work on a project, write an e-mail to his mother, and (play, to play) in a softball game. 2. Three of the events in a decathlon are the 100-meter dash, (pole vaulting, pole vault), and long jump. 3. Listening to a recording of your favorite band is not quite the same as (to listen, listening) to that band at a live concert. 4. The mainland was visited by explorers much later than (the outlying islands, were the outlying islands).

IDIOMS Idioms are fixed expressions, groups of words that are used together. There’s really no rule that applies to these expressions—the conventions of English simply demand that they be phrased a certain way. Idiom errors show up frequently in sentence correction questions, often as a secondary error. ETS seems to feel that idioms are critical to determining your ability to pursue a graduate business degree, so you need to be concerned with these expressions.

Quick Quiz: Idioms Fill in the missing word in each sentence. 1. She is not only beautiful, ____________ smart. 2. I can’t distinguish day ____________ night. 3. I can distinguish between black ____________ white. 4. My GMAT teacher defines the conclusion ____________ the main point of the argument. 5. If you take the GMAT enough times, you might develop the ability ____________ choose the credited responses without reading the questions. 6. Art historians regard the Mona Lisa ____________ one of the greatest works of art. 7. Art historians consider the Mona Lisa ____________ one of the greatest works of art. 8. He is not so much smart ____________ cunning. 9. The mule, ____________ the donkey, is a close relative of the horse. 10. Many of my favorite ice cream flavors, ____________ chocolate chip and strawberry, are also available as frozen yogurt. 11. Her coat is just ____________ mine. 12. He walks to work, just ____________ I do. How did you do? Check the answers at the end of the chapter to see how well you know your idioms. As you may have noticed from the quiz, idioms often involve prepositions, those tiny words that establish the relationship between other words.

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Assignment 2

The best way to learn idioms is through repetition and memorization. Don’t get hung up on the “why.” Accept that, as former news anchorman Walter Cronkite used to put it, “that’s the way it is.” We’ve made the process a little easier by providing a list of the idioms commonly tested on the GMAT. Review the idiom list below, and note any idioms that give you difficulty. Spend time learning those expressions.

Idiom List The following list contains the idioms tested most frequently on the GMAT: ABOUT Worry...about If you worry too much about the GMAT, you’ll develop an ulcer. AS Define...as My GMAT teacher defines the conclusion as the main point of the argument. Regard...as Art historians regard the Mona Lisa as one of the greatest works of art. Not so...as He is not so much smart as cunning. So...as to be She is so beautiful as to be exquisite. Think of...as Think of it more as a promise than a threat. See...as Many people see euthanasia as an escape from pain. The same...as Mom and Dad gave the same punishment to me as to you. As...as Memorizing idioms is not as fun as playing bingo. AT Target...at The commercials were obviously targeted at teenage boys. FOR Responsible for You are responsible for the child.

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FROM Prohibit...from He was prohibited from entering the public library after he accidentally set the dictionary on fire with a magnifying glass. Different...from Democrats are not so different from Republicans in the United States. OVER Dispute over The men had a dispute over money. THAT So... that He was so late that he missed the main course. Hypothesis...that The hypothesis that aspartame causes brain tumors has not been proven yet. TO BE Believe...to be His friends do not believe the ring he bought at the auction to be Jackie O’s; they all think he was tricked. Estimate...to be The time he has spent impersonating Elvis is estimated to be longer than the time Elvis himself spent performing. TO Forbid...to I forbid you to call me before noon. Ability...to If you took the GMAT enough times, you might develop the ability to choose the credited responses without reading the questions. Attribute...to Many amusing quips are attributed to Dorothy Parker. Require...to Before you enter the house you are required to take off your hat. Responsibility to You have a responsibility to take care of the child. Permit...to I don’t permit my children to play with knives in the living room. Superior...to My pasta sauce is far superior to my mother-in-law’s.

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Assignment 2

Try...to Try to stay awake during the essay section of the test. WITH Credit...with Many people credit Christopher Columbus with the discovery of America, but Native Americans were here first. Associate...with Most politicians prefer not to be associated with the Mafia. Contrast...with My father likes to contrast my grades with my brother’s. NO PREPOSITION Consider...(nothing) Art historians consider the Mona Lisa one of the greatest works of art. MORE THAN ONE PREPOSITION Distinguish...from I can’t distinguish day from night. Distinguish between...and I can distinguish between black and white. Native (noun)... of Mel Gibson is a native of Australia. Native (adjective)…to The kangaroo is native to Australia. COMPARISONS AND LINKS Not only...but also She is not only beautiful, but also smart. Not...but The review was not mean-spirited but merely flippant. Either...or I must have either chocolate ice cream or carrot cake to complete a great meal. Neither...nor Because Jenny was grounded, she could neither leave the house nor use the telephone. Both...and When given the choice, I choose both ice cream and cake. More...than; Less...than The chimpanzee is much more intelligent than the orangutan.

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GMAT MANUAL

As vs. like As is used to compare actions. Like is used to compare nouns. He did not vote for the Libertarian Party, as I did. Her coat is just like mine. Like vs. such as Like means similar to. Such as means for example. The mule, like the donkey, is a close relative of the horse. Many of my favorite ice cream flavors, such as chocolate chip and strawberry, are also available as frozen yogurt. The more...the -er The more you ignore me, the closer I get. From...to Scores on the GMAT range from 200 to 800. Just as...so too Just as I crossed over to the dark side, so too will you, my son. MISCELLANEOUS Each vs. all or both Use each when you want to emphasize the separateness of the items. Use both (for two things) or all (for more than two things) when you want to emphasize the togetherness of the items. Each of the doctors had his own specialty. Both of the women went to Bryn Mawr for their undergraduate degrees. All of the letters received before January 15 went into the drawing for the $10 million prize. Whether vs. if Use whether when there are two possibilities. Use if in conditional statements. Eduardo wasn’t sure whether he could make it to the party. If Eduardo comes to the party, he will bring a bottle of wine.

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Assignment 2

SUMMARY Modifiers • Modifiers describe, or modify, other words in a sentence. Adjectives modify nouns. Adverbs modify verbs, adjectives, and other adverbs. • Modifiers must go next to what they modify; otherwise, the modifier has been misplaced. • To choose the correct quantity word, decide if the item is countable (such as pencils, coins, or stock options) or not countable (such as Jell-O, love, or soup.) • Use fewer, number, and many to describe countable nouns. Use less, amount, quantity, and much to describe non-countable nouns. • Use between and -er adjectives to compare two things and among and –est adjectives to compare three or more things.

Parallel Construction • Parallel construction is required for lists and comparisons. Each item in the list or comparison must have the same grammatical construction.

Idioms • Idioms are fixed expressions. • Study the idiom list to learn the idioms tested on the GMAT.

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ANSWERS AND EXPLANATIONS Quick Quiz: Modifiers 1. Walking down the avenue, I was caught in a torrential downpour. Walking down the avenue modifies I. Torrential modifies downpour. 2. Left in the refrigerator for several weeks, the meat was now spoiled. Left in the refrigerator for several weeks modifies meat. Spoiled also modifies meat. 3. Michael Jordan, who is now 40 years old, is still considered one of the best players in professional basketball. Who is now 40 years old modifies Michael Jordan. Still modifies considered. Best modifies players. Professional modifies basketball.

Quick Quiz: Misplaced Modifiers 1. Arranged in secret, the discovery of Romeo and Juliet’s marriage was made only after their deaths. Misplaced modifier. The phrase arranged in secret modifies discovery, but it was really the marriage that was arranged secretly. The corrected sentence: Arranged in secret, Romeo and Juliet’s marriage was discovered only after their deaths. 2. Discovered by Marie Curie and Pierre Curie, polonium and radium were first isolated in 1898. No error. The phrase Discovered by Marie Curie and Pierre Curie correctly modifies polonium and radium. 3. Invented by James Hargreaves in 1765, the spinning jenny was capable of spinning eight to eleven threads at one time. No error. The phrase Invented by James Hargreaves in 1765 correctly modifies the spinning jenny. 4. I overheard him say that he had cheated on the exam while I was standing in the hallway. Misplaced modifier. Because the clause while I was standing in the hallway is at the end of the sentence, it seems to describe when the cheating occurred, not when the conversation was overheard. The corrected sentence: While I was standing in the hallway, I overheard him say that he had cheated on the exam. 5. Once a very powerful nation, France’s status has declined in recent years. Misplaced modifier. The phrase Once a very powerful nation modifies status, but it should modify France. The corrected sentence: Once a very powerful nation, France has declined in status in recent years.

Quick Quiz: Quantity Words 1. The better you prepare for the GMAT, the higher your score will be. 2. Most of the population lives in poverty. 3. Many of the people live in poverty. 4. Since I withdrew money from my bank account, the number of dollars in the account is now fewer. 5. Since I withdrew money from my bank account, the amount of money in the account is now less. 6. Some people consider the Yankees to be the greatest baseball team ever.

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Assignment 2

Quick Quiz: Parallelism 1. On Saturday, David had to work on a project, write an e-mail to his mother, and play in a softball game. 2. Three of the events in a decathlon are the 100-meter dash, pole vault, and long jump. 3. Listening to a recording of your favorite band is not quite the same as listening to that band at a live concert. 4. The mainland was visited by explorers much later than were the outlying islands.

Quick Quiz: Idioms The idiom is in bold text, and the word that goes in the blank is underlined. 1. She is not only beautiful, but also smart. The idiom is not only...but also. 2. I can’t distinguish day from night. The idiom is distinguish x from y. 3. I can distinguish between black and white. The idiom is distinguish between x and y. Note that there are two idioms for distinguish. You can use either one, but you can’t mix them up. For example, “I can’t distinguish day and night,” and “I can’t distinguish between day from night,” are wrong. 4. My GMAT teacher defines the conclusion as the main point of the argument. The idiom is define x as y. 5. If you take the GMAT enough times, you might develop the ability to choose the credited responses without reading the questions. The idiom is ability to. 6. Art historians regard the Mona Lisa as one of the greatest works of art. The idiom is regard x as y. 7. Art historians consider the Mona Lisa one of the greatest works of art. Trick question! The idiom is consider, and it should not be followed by any prepositions. Though we often say “consider to be” in everyday speech, the GMAT writers consider this phrase incorrect. 8. He is not so much smart as cunning. The idiom is not so much x as y. 9. The mule, like the donkey, is a close relative of the horse. Use like when you mean “is similar to.” 10. Many of my favorite ice cream flavors, such as chocolate chip and strawberry, are also available as frozen yogurt. Use such as when you mean “for example.” 11. Her coat is just like mine. This sentence compares her coat and my coat. Use like to compare nouns. 12. He walks to work, just as I do. This sentence compares the action of walking. Use as to compare verbs.

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PART-TO-WHOLE RELATIONSHIPS Fractions, decimals, and percents are three ways to express a part of a whole. Many questions on the GMAT require you to perform basic operations (adding, subtracting, multiplying, and dividing) with fractions, decimals, and percents.

FRACTIONS Here’s a quick review of fractions: • A fraction describes a

part . whole

• The top is the numerator. The bottom is the denominator. For example, in the fraction

5 , 5 is the numerator and 7 is the 7

denominator. • The reciprocal (also called the inverse) of a fraction is that fraction flipped over. The reciprocal of

5 7 is . 7 5

• The fraction bar (the line between the numerator and the denominator) is equivalent to division. For example,

10 means 5

10 ÷ 5, or 2. Let’s take a look at how ETS might test your knowledge of fractions.

 3 2  +   2 3  1. =  1 1  ×   4 5 

1 20 5 20 13 6 20 130 3 First, let’s deal with adding 3 and 2 . 2 3

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Assignment 2

If fractions have the same denominator, you simply add or subtract the numerators and put the sum over the denominator. Don’t change the denominator. You may recall learning something about finding the lowest common denominator to add or subtract fractions with different denominators. Well, we here at the Princeton Review have something much cooler. We call it the Bowtie. Here’s how it works:

Use the Bowtie method to add or subtract fractions.

First, multiply diagonally up (opposing denominators and numerators). 9 3 2

4 +

2 3

Second, carry up the sign (in this case, addition). 9

+

4

3 2

+

2 = 3

Third, add (or subtract) across the top. 9

+

4

3 2

+

2 13 = 3

Finally, multiply across the bottom. 9

+

4

3 2

+

2 13 = 3 6

Now, our original problem is:  3 2 13  +   2 3 = 6  1 1 1 1 ×  ×  4 5  4 5

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GMAT MANUAL

To multiply fractions, multiply straight across the top and the bottom.

1 4

×

1

=

5

1× 1 4 ×5

=

1 20

Now, our problem is: 13 6 1 20 Rewritten, this is 13 ÷ 1 . 6 20 To divide fractions, multiply by the reciprocal of the second fraction: 13 6

÷

1 20

=

13 6

×

20 1

=

13 × 20 6 ×1

=

260 6

To reduce a fraction, divide the top and bottom by the same number. Both the top and the bottom can be divided by 2: 260 ÷ 2 If possible, reduce before you multiply.

6÷2

=

130 3

Try to reduce before you multiply. When multiplying fractions, you can divide either numerator or either denominator by the same number to reduce. Do NOT do this with addition or subtraction problems. Let’s look at that problem again. We can divide a top number (20) by 2, and a bottom number (6) by 2:

10 13 × 6 3

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20 1

= 130 3

Assignment 2

Quick Quiz: Fractions 1.

2.

3.

4.

5.

3 4 2 5 4 3 3 5 5 6

1

÷

+

7

=

1 13

–

5 2

=

=

×

15

−

1

2

14

=

=

Try this example: 2.

2 3

+

2 9

+

2 27

+

2 81

= 40 81 68 81 80 81 1 107 81

Would you use the Bowtie in this problem? Well, as handy as the Bowtie is, every once in a while it’s easier to come up with the lowest common denominator. Use the lowest common denominator when all the numbers in the denominators are factors or multiples of one another. What number would serve as a common denominator for all of these fractions? ______

The Bowtie method isn’t always the fastest way to add fractions.

To put each of the fractions over the same common denominator, decide what number you need to multiply the denominator by in order to convert it to the lowest common denominator. Then multiply the top and bottom of each fraction 2 by that number. For instance, to convert : 3

2 × 27 54 = 3 × 27 81 © Princeton Review Management, L. L. C.

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Now, put the other fractions over 81 and then add them up 54 18 6 2 80 + + + = ). The correct answer is therefore (C). ( 81 81 81 81 81

Fractions Between Zero and One Let’s consider what happens when you perform the basic operations with numbers between 0 and 1. Fill in the answer, then circle either “bigger” or “smaller.” 1

1

+

3

= ______

bigger/smaller

2

Given any number, if you add to it a number between 0 and 1, the result will be bigger than the original number. 1

–

3

1

= ______

bigger/smaller

2

Given any number, if you subtract from it a number between 0 and 1, the result will be smaller than the original number. 1 3

×

1

= ______

bigger/smaller

2

Given a positive number, if you multiply it by a number between 0 and 1, the result will be smaller than the original number. 1 3

÷

1

= ______

bigger/smaller

2

Given a positive number, if you divide it by a number between 0 and 1, the result will be bigger than the original number.

DECIMALS Decimals and fractions go hand in hand. On the GMAT, every decimal can be written as a fraction, and every fraction can be written as a decimal. Do your calculations in the format with which you’re most comfortable.

0.5 =

5 10

3 5

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=

1 2

= 3 ÷ 5 = 0.6

Assignment 2

To convert a decimal to a fraction, place the decimal over 1, move the decimal points the same number of places on the top and bottom until you have whole numbers on both the top and the bottom, then reduce.

0.25 =

0.25

25

=

1.00

=

100

1 4

Convert these to fractions: 0.4 = ______

0.125 = ______

To convert a fraction to a decimal, divide the bottom into the top.

1

)

.25

= 4 1.00 = 0.25

4

Convert these to decimals: 3

= ______

8

7

= ______

20

Now, let’s take a look at a question: 1. 3

1

– 3.025 =

4 0 0.025 0.125 0.225 0.25 Before we subtract, we need to convert the fraction to a decimal: 1 4

)

.25

= 4 1.00 = 0.25, so 3

1

= 3.25.

4

Whenever you add or subtract decimals, always line up the decimal points. You can add zeros onto the end of a decimal without changing value. 3.250 –3.025

Line up the decimal points when you add or subtract.

0.225 The answer is (D).

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GMAT MANUAL

Here’s another problem: 2. (0.2)(0.3)(0.05) = 0.0003 0.003 0.03 0.3 3.0 Here’s a good way to multiply decimals. First, ignore the decimal points and simply multiply the numbers: 2 × 3 × 5 = 30. Second, count the number of decimal places in each of the original decimals: 0.2 and 0.3 each have 1 decimal place and 0.05 has 2 decimal places, which gives a total of 4 decimal places. Third, take your product and move the decimal the same number of places to the left: 0.0030

Therefore, the answer is (B). Here’s another decimal example:

3. Which of the following is the closest to

5.13

?

0.02 2500 250 25 2.5 0.25 When you divide by a decimal, you can simply change your decimal into a whole number. To do this, move the decimal point in both your numerator and your denominator the same number of places to the right. You can only do this when you divide decimals! 5.13 513 = 0.02 2 Notice that the question is not asking for the exact solution. You only need to find an approximation. 513 is about 500. 500 divided by 2 is 250. Thus, the answer is (B).

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Assignment 2

DIGIT PLACES Some questions on the GMAT involve digit places. Consider the number 6,493.783. 6 , 4

9

thousands hundreds tens

3 . 7

8

3 thousandths

hundredths tenths units

Each place contains a digit, which is a whole number from 0 to 9. For example, 7,542 is a four-digit number. Round decimals by looking at the digit to the right of the decimal place to which you want to round. For example, to round to the nearest hundredth, look at the thousandths place. If the thousandths place is 5 or more, round the hundredths place up by 1. For example, 0.247 rounded to the nearest hundredth is 0.25. If the thousandths place is less than 5, the hundredths place stays the same. For example, 0.241 rounded to the nearest hundredth is 0.24.

Quick Quiz: Decimals 1. What is 56.7189 rounded to the nearest tenth? 2. What is 1.1119 rounded to the nearest thousandth? 3. 43.25 + 3.9 = 4. 2.1 × 0.002 = 5. 3.02 ÷ 0.2 =

PERCENTS Percents, like fractions and decimals, are just another way of describing a partto-whole relationship. Any percent can also be written as either a fraction or a decimal. “Percent” simply means “over one hundred.” Any percent can be expressed as a fraction by putting it over 100, then reducing the fraction. For example,

3% =

3 100

25% =

25 100

=

1 4

To convert a fraction into a percent, you have to know that part = x , where whole 100 x is the percentage. For instance, 3 is the same as 60 or 60%. 5 100

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To convert a percent to a decimal, drop the percent sign and move the decimal point two places to the left: 25

25% =

= 0.25

5

5% =

100 0.1% =

= 0.05

100

.1

= 0.001

200

200% =

100

= 2.0

100

Likewise, to convert a decimal to a percent, move the decimal point two places to the right and add a percent sign:

0.18 =

18

= 18%

100 3.5 =

350

70

0.7 =

= 70%

100

= 350%

0.002 =

.2

= 0.2%

100

100

And now for the trick question: Convert 0.003 into a percent. That’s right, it’s 0.3%. It is NOT 3%. Note that percents can be less than 1. Be careful with your place holders. As you may have noticed, ETS likes to phrase percent problems on the GMAT as long word problems. Instead of trying to invent your own equation for the problem, use this table to translate a percent word problem into a mathematical equation.

English

Mathematical translation

percent

100

of

× (times)

what

x, y, or z (a variable)

is, are, were

=

Try using translation on this problem before reading the explanation below it. 1. 6.5 is what percent of 1,300? 0.5% 1% 5% 50% 500% 6.5

is

6.5

=

what percent

of

1,300

y

×

1,300

100

Then, solve for y. The correct answer is (A).

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Assignment 2

Let’s try a word problem: 2. A merchant raises the price of a $100 item by 20%. After finding that she cannot sell the item at the higher price, she discounts it by 20%. What is the final price of the item? $96 $97 $98 $99 $100 Take it one sentence at a time. First, we need to increase $100 by 20%. One way to do this is to find 20% of $100 and then add that amount. 20% of $100 is $20. So, $100 + $20 = $120. Be careful. The problem says that the higher price (not the original price) was discounted by 20%. We need to find 20% of $120. 20 1 × $120 = × $120 = $24. The final price is $120 – $24 = $96. The answer is (A). 100 5

Percent Change Here’s a slightly different kind of percent problem: 3. The average television prime-time advertising unit of 30 seconds cost $30,000 in 1973 and $50,000 in 1977. What was the approximate percent increase in the cost of a unit? 20% 33% 40% 60% 67% Many percent questions ask for the percent increase or the percent decrease from one number to another. For such questions, use this formula:

Percent change =

difference × 100 original

The difference is what you get when you subtract the smaller number from the larger number: 50,000 – 30,000 = 20,000. The original is the starting number. Putting these into our formula, we get:

Percent change =

20, 000 30, 000

2 3

× 100 =

200 3

% = 66

2

× 100

%, which is closest to answer choice (E).

3

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GMAT MANUAL

Deciding which number counts as the “original” can be confusing. Remember, if the question asks for percent increase, the original is the smaller number. If the question asks for percent decrease, the original is the larger number.

Shortcuts to Calculate Percents Always looking for the shortcut, eh? That’s wise on a timed test. Here’s a shortcut for percent problems: First, it’s easy to find 10% or 1% of a number: just move the decimal point one or two places to the left. For instance, 10% of 1,024 is 102.4. Once you have 10%, it’s easy to find 5% (by dividing your result by 2) or 30% (by multiplying your result by 3), etc. Thus, 5% of 1,024 is 51.2. 30% of 1,024 is 307.2. You can even use this GMAT shortcut in real life: To calculate a 15% tip on a $156 dinner tab, you can figure out that 10% + 5% = $15.60 + $7.80 = $23.40. Want a further shortcut? Before you calculate, eyeball the answer choices to see how exact you need to be in your calculations. For example, if the answer choices aren’t terribly close in value, saying 5% of 1,024 is “a bit more than 50” will do just fine.

Comparing Decimals, Fractions, and Percents It’s worth memorizing this chart. It will save you a lot of time on the GMAT if you can comfortably switch among decimals, fractions, and percents.

Fraction 1 2 1 3 2 3 1 4 3 4 1 5 2 5 3 5 4 5 1 6 1 8

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Decimal

Percent

0.5

50%

0.33

33.3%

0.66

66.6%

0.25

25%

0.75

75%

0.2

20%

0.4

40%

0.6

60%

0.8

80%

0.166

16.6%

0.125

12.5%

Assignment 2

Did you memorize the numbers above, or did you just scan over them? Which ones did you already know? Which ones don’t you know YET? Try converting some of the ones you don’t know yet, then check the chart. For example: What’s 1 as a decimal? What’s 40% as a fraction? 6

Estimating Percents As you know, your job on the GMAT isn’t to calculate the correct answer. It’s to do just enough on the problem to spot which answer choice is correct. Often, you just need to estimate the correct answer. For example, pretend that you just worked a long problem. You’ve gotten to the end of it and the last thing to do is: 21120 64000 Yuck. Before you calculate this, look at the answer choices: 22% 33% 44% 66% 77% Notice that the answer choices are not terribly close in value. Thus, you just need a rough calculation of your fraction: 21120 64000

≈

20000 60000

=

2 6

=

1

1 = 33 %. Therefore, the answer is (B). 3 3

Quick Quiz: Estimating Come up with rough estimates for the following: 1. 32% of 6,050 is approximately ______. 2. 9.2% of 41 is approximately ______. 3.

374

is approximately ______%.

720 4.

50

is approximately ______%.

2412

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PROBABILITY You’ve heard probability used in a variety of ways. There’s a 30% chance of rain today. There’s a one in a million chance of beating Michael Jordan at a game of HORSE. Probabilities are just another type of part-to-whole relationship. They can be expressed as fractions, decimals, or percents. Probability is always between zero and one, inclusive. If something can absolutely never happen, its probability is 0. If something absolutely will happen, it’s probability is 1 (or 100%). This is handy to keep in mind for ballparking. If 1 the situation seems unlikely, then its probability is less than . If it seems likely, 2 1 then its probability is greater than . 2 The formula is:

probability =

number of outcomes you want number of total possible outcomes

Let’s look at an example. 1. Jeffrey has a bag of marbles. The bag contains 6 red, 6 yellow, 12 green, and 12 blue marbles. It contains no other marbles. What is the probability that a marble chosen at random will be either red or yellow? 1 6 1 3 1 2 2 3 3 4 First, find the number of outcomes you want. This means the number of ways that you could get a red or yellow marble: 6 + 6 = 12. Next, find the number of total possible outcomes. The total outcomes are red, yellow, green, and blue: 6 + 6 + 12 + 12 = 36. Now, put it in the formula. The probability of getting a red or yellow = 12 = 1 . So, the answer is (B). 36 3 66

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Assignment 2

Quick Quiz: Probability 1 Jose has a box of doughnuts. There are 4 plain doughnuts, 2 jelly doughnuts, and 8 chocolate doughnuts. If he picks one at random, what is the probability that he will pick a chocolate one? 2. A streetlight is green for 15 seconds, yellow for 5 seconds, and red for 30 seconds. What is the probability that the light will be red the moment you arrive at the intersection? 3. Mary made trail mix that has 30 peanuts, 40 raisins, 10 chocolate candies, and 15 banana chips. She picked one item from the bag at random. What is the probability that she picked either a chocolate candy or a banana chip?

INTEREST GMAT problems may ask you to calculate simple or compound interest. Simple interest is as easy as finding a percentage. Here’s an example: 1. If Molly puts $500 in a savings account that pays 4 percent simple annual interest, how much money will be in the account after one year? $20 $500 $504 $520 $540 This is a basic percentage problem: What is 4 percent of $500?

y=

4 100

× 500

y = 20 The account earns $20 over the course of the year, so there would be $520 in the account. The answer is (D). Compound interest is a slightly different animal. It means that you earn interest on your interest. Really, you only end up earning slightly more than you would have earned with simple interest. Compound interest problems are fairly rare on the GMAT. When they do show up, you can usually just calculate the simple interest, then look for the answer choice that is slightly more than that. For example: 2. If Molly puts $500 in a savings account that pays 4 percent annual interest compounded semiannually, how much money will be in the account after one year? $5.20 $504.00 $520.00 $520.20 $540.00

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GMAT MANUAL

We calculated that, with simple interest there would have been $520 in the bank. So, we need the answer choice that is slightly more than $520. That would be (D), $520.20. Dying for the formula? Here it is. You might see a problem in which you need to recognize the formula, but you won’t have to calculate it out.

principal + interest = principal × (1 + r)t r = interest rate for the compounding period expressed as a decimal t = number of compounding periods So, for Molly: Use r = 0.02 because she earns 2 percent per half year. Use t = 2 because one year contains two compounding periods. Thus principal + interest = 500 × (1 + 0.02)2 = 520.20. So, the answer is still (D).

Quick Quiz: Interest 1. Kevin has $1,500 in a bank account that pays 5 percent simple interest annually. How much money will be in the account after one year? 2. Sandy has $2,000 in a bank account that pays 4 percent interest compounded annually. After three years, approximately how much money will be in the account?

DATA SUFFICIENCY RANGES To answer a question asking the value of something, you need to be able to find a single value. If Statements (1) and (2) each give a list of possible values, and the two lists have exactly one value in common, then the correct answer is (C). Let’s look at an example: 1. What is the value of the odd integer x? (1) 18 < 2x < 30 (2) 33 < 3x < 75 If you divide the expression in Statement (1) by 2, the result is 9 < x < 15. So x (an odd integer) can be 11 or 13. Write BCE. Fact (2) can be divided by 3 to give 11 < x < 25. So x = 13, 15, 17, 19, 21, or 23. Eliminate (B). If you combine (1) and (2), the only number that makes both statements true is 13. Thus, there is a single value for x, and the answer is (C). 2. What is the value of x? (1) x = 4 (2) x 2 – 16 = 0

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Assignment 2

Since both 4 and –4 have an absolute value of 4, Statement (1) is insufficient. Write BCE. Similarly, both 4 and –4 work for Statement (2). Eliminate (B). Even when you combine (1) and (2), you still have two possible values. Since you cannot determine a single value for x, the answer is (E).

SUMMARY Fractions • Terms: The top is the numerator. The bottom is the denominator. The reciprocal is the fraction flipped over. • The fraction bar is equivalent to ÷. • To add or subtract fractions, use the Bowtie. • To multiply fractions, multiply straight across. • To divide fractions, multiply by the reciprocal of the second fraction.

Decimals • Adding and subtracting decimals: Always line up the decimal points. • Multiplying decimals: Ignore the decimal points and multiply the numbers, count the number of decimal places in each of the original decimals, then move the decimal point the same number of places to the left. • Dividing by a decimal: Move the decimal point in both the numerator and the denominator the same number of places to the right. You can only do this when you divide decimals!

Percents • “Percent” means “over 100.” • Use translation chart for wordy percent problems. • Percent change =

difference × 100 original

• “Percent increase” tells you that the original is the smaller number. “Percent decrease” tells you that the original is the larger number. • Whenever possible, estimate percents.

Converting Fractions, Decimals, and Percents • To convert a decimal to a fraction, place the decimal over 1, move the decimal points to the right the same number of places on the top and bottom, then reduce. • To convert a percent to a fraction: Put the percent over 100, and reduce the fraction. • To convert a fraction into a percent,

part x = , where x is the whole 100

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GMAT MANUAL

• To convert a percent to a decimal, drop the percent sign and move the decimal point two places to the left. • To convert a decimal to a percent, move the decimal point two places to the right and add a percent sign.

Probability • Probability is always between zero and one inclusive. •

probability =

number of outcomes you want number of total possible outcomes

Interest • For simple interest, find the percent. • For compound interest, find the percent, then look for the answer choice slightly more than this result.

DRILL 1. Franklin’s wage increased by what percent? (1) Franklin’s wage after the increase was $45,000. (2) Franklin’s wage increased by $2,000. 2. Which of the following is the greatest? 0.03 1 – 0.3 0.3 3 1 0.3 0.3 3. The profits from the three divisions of Company X totaled $53,000. The profit from Division A was $26,000. The profit from Division B was 20 percent of the profit that was not from Division A. The rest of the profit came from Division C. What was the amount of profit from Division C? $5,400 $20,800 $21,600 $27,000 $42,400

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Assignment 2

4. What is the one hundred-thousandths digit in the decimal form of

3

?

11 0 1 2 6 7

5.

 1 2 1 + 3  5  3 2 1  +2  3 5 1

30 41

2 2 4

11 15 11

15 46 10 225 6. Company XYZ had $498.2 million in profits for the year. The sales department brought in $302.6 million of the profits. Approximately what percent of the profits were not brought in by the sales department? 4% 6% 25% 40% 60% 7. In 1998, Company Q had $359,000 in profits. In 2002, Company Q’s profits were 250 percent greater than they were in 1998. Approximately what were Company Q’s profits in 2002? $144,000 $450,000 $610,000 $900,000 $1,260,000 8. A total of 20 interns and 15 full-time employees work in the 35 offices of a company, with one person working in each office. One office is selected at random to get an upgraded desk. What is the probability that the office selected will be that of a female intern? (1) Of the interns, 15 of them are female. (2) Of the interns, 5 of them are male.

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GMAT MANUAL

Given a positive number, if you divide it by a number between 0 and 1, the result will be bigger than the original number.

ANSWERS AND EXPLANATIONS Quick Quiz: Fractions 1. (

21 ) Flip the fraction, then multiply straight across: 4

Convert these to fractions:

3 7 3 × 7 21 × = = 4 1 4 ×1 4

2. (

Decimals

0.4 =

2 5

0.125 =

1 8

31 26 + 5 31 ) Use the Bowtie: = 65 5 × 13 65

Quick Quiz: Decimals 3. (

7 8 − 15 ) Use the Bowtie: = 6 3×2

7 6

1. 56.7 2. 1.112

4. (

9 3 15 3 3 ) Reduce the 5 and the 15 by 5: × = × , 2 5 2 1 2

then multiply straight across:

5. (

3×3 9 = 1×2 2

16 70 − 6 64 ) Use the Bowtie: = . 21 6 × 14 84

Then reduce by 4:

64 16 = 84 21

3. 47.15 4. .0042 5. 15.1

Quick Quiz: Estimating 1 1. Approximately 2,000. 32% is close to 33 %, which 3 1 1 is . 6,050 is close to 6,000. of 6,000 is 2,000. 3 3

Fractions Between Zero and One 2. Approximately 4. 9.2% is close to 10%. 41 is close to 1 1 5 + = 3 2 6

bigger

Given any number, if you add to it a number between 0 and 1, the result will be bigger than the original number. 1 1 1 − =− 3 2 6

smaller

Given any number, if you subtract from it a number between 0 and 1, the result will be smaller than the original number.

40. 10% of 40 is 4. 3. Approximately 50%.

374 350 1 is close to = = 50%. 720 700 2

4. Approximately 2%.

50 50 1 is close to = = 2%. 2412 2500 50

Quick Quiz: Probability 1. (

1 1 1 × = 3 2 6

smaller

Given a positive number, if you multiply it by a number between 0 and 1, the result will be smaller than the original number. 1 1 2 ÷ = 3 2 3 72

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bigger

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4 ) The “number of outcomes you want” is “8 7 chocolate doughnuts.” Divide this by the total: 8 4 4 + 2 + 8 = 14. Thus, . Reduce to . 14 7

Assignment 2

2. (

3. (

3 ) The “number of outcomes you want” 5 is “red for 30 seconds.” Divide this 30 by the total: 15 + 5 + 30. Thus, . 50 3 Reduce to . 5

Drill 1. C

5 ) The “number of outcomes you want” 19 is “10 chocolate candies or 15 banana chips”: 10 chocolate candies + 15 banana chips = 25. Divide this by the 25 total: 30 + 40 + 10 + 15 = 95. Thus, . 95 5 Reduce to . 19

Quick Quiz: Interest 5 × 1,500 100 = 75. Then, add that to the 1,500 that’s in the bank: $75 + $1,500 = $1,575.

1. $1,575 Find 5 percent of $1,500:

2.

2. D Ballpark the answer choices to see if they’re more than 1 or less than 1. (A), (B), and (E) are less than one. Take a moment with (C) and (D). (C) is 0.1. (D) is more than one, because when you divide by a number between 0 and 1, the number gets bigger. So, only (D) is greater than 1. It must be the greatest. 3. C

First, find the profit that is “not from Division A”: $53,000 – $26,000 = $27,000. Division B is responsible for

Slightly more than $2,240. Find the

20% of this. That means that Division

simple interest and ballpark. The 4 simple interest would be × $2,000 100 = $80. She made this for three years,

C is responsible for the remaining 80%. Translate: Division C is 80% of the profit not from Division A. C = 80 × 100 $27,000 = $21,600.

so she made $80 × 3 = $240. Add this to Sandy’s $2,000 in the bank, and you get $2,240. She would have $2,240 if

In order to know the percent increase, we need to be able to find out the difference and the original. Statement (1) does not give us a way to find the difference or the original. So, cross out (A) and (D). Statement (2) only tells us the difference. Cross out (B). Together, we have the difference (from the second statement) and a way to find the original ($45,000 – $2,000). Thus, the statements together are sufficient to answer the question.

the compound interest is slightly

Divide out until you see a pattern: 0.27272 11 3.00000 . The one hundredthousandths place is five places to the

more than this. If you’re just itching

right of the decimal point. The order

to know, the precise answer (if you

is: tenths, hundredths, thousandths,

it was simple interest, so you know

used the formula) is $2,249.73.

4. C

)

ten-thousandths, one hundredthousandths.

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5. A

Make your life easier by converting 1 these into improper fractions. ( 1 + 3 4 17 2 3 ) = ( + ). Then bowtie: 3 5 5 20 + 51 = 71 . Do the same for the 3×5 15 41 6 + 35 2 7 denominator: ( + ) = ( )= . 15 3×5 5 3

So, the problem is now: 71 41 71 15 ÷ = – . Take 15 out of the 15 15 15 41 71 1 71 30 top and bottom: – = =1 . 1 41 41 41

6. D Two important words to notice in the question: “approximately” and “not.” “Approximately” is a tip-off to use nicer numbers, such as $500 (million) and $300 (million). The profits NOT brought in from the sales department were: 500 – 300 = 200. Then, translate “what percent of the profits were not brought in by the sales department”:

y × 500 = 200, which yields y = 40. 100 7. E

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Notice that it’s “250 percent greater,” NOT “250 percent of.” Also notice the “approximately” that tells you it’s okay to use a nicer number, such as $360,000. You can use the percent change formula on this one: percent change = (Difference ÷ original) × 100. So, 250% = (Difference ÷ 360,000) × 100. Solve for Difference = $900,000. YOU’RE NOT DONE! That’s the difference between the years. Add this to the original, and you’ll get the profits for 2002 (which is what the question is asking for): $360,000 + $900,000 = $1,260,000.

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8. D To know the probability, we need the “number of outcomes we want,” which is the number of female interns and the total number of possibilities. The question stem gives us the total possibilities: 35 offices. Thus, all we need from the statements is some way to find the number of female interns. Statement (1) gives us that info. Cross out (B), (C), and (E). Statement (2) also gives us that information (assuming the only options are male and female, we can subtract the male interns from the total interns to get the number of female interns).

Assignment 2

ADMISSIONS INSIGHT Application Timeline Here’s the business school game plan: January–May Research schools. Take GMAT prep course and study hard. June–July Take the GMAT. Request official undergraduate transcripts. Consider a business school admissions counseling service. Secure recommenders. July–August Complete and submit GMAC Additional Score Reporting (ASR) form if needed. Consider applying online. Start those essays. Follow up with recommenders. Update resume. September Fine-tune essays. Stay in touch with recommenders; make sure they can meet deadlines. October–November Submit first applications. Send thank-you notes to recommenders. November–December Submit more applications. January–March Start planning for fall.

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ASSIGNMENT 3

ARGUMENTS Arguments is The Princeton Review’s name for the critical reasoning questions. Arguments questions typically make up about 30 percent of the Verbal section. You can expect to see about twelve arguments questions on your exam. Each question is composed of a short passage, a question about that passage, and five answer choices. With the Princeton Review’s approach to arguments, you will learn how to analyze the reasoning in an argument, identify the different types of arguments questions, and determine the answer by using Process of Elimination (POE). This lesson introduces some key skills needed to solve arguments questions: identifying argument components and recognizing common reasoning errors.

PARTS OF AN ARGUMENT Becoming an active reader will help you work arguments questions efficiently and accurately. For most questions on the GMAT, begin by breaking the argument down into its parts. Three connected parts make up an argument. The first two, the conclusion and premises, are stated explicitly in the argument, while the third part, the assumption, is unwritten. Let’s look at each of the parts in more detail.

Conclusion The conclusion is the main point or central claim of the argument. Think of an argument as a television commercial. After you read the argument, ask yourself, “What is the author trying to sell me?” The claim the author wants you to accept is the conclusion of the argument. Often, you will see indicator words that will help you find the conclusion. Common conclusion indicator words include: • Therefore

• Hence

• Clearly

• Consequently

• Thus

• In conclusion

• So

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Premises Once you have found the conclusion, identify the premises—any reasons, statistics, or evidence—the author provides to support the conclusion. Premises usually sound like facts, rather than opinions. Even if you disagree with the premises provided as support for a conclusion, you must accept them as true for the purposes of the GMAT. Sometimes indicator words can help you find the premises. Look for words like: • since

• as a result of

• because

• suppose

The Why Test Indicator words can help you find the conclusion and premises, but not every argument uses them. The most reliable method for identifying these parts is the Why Test. Once you have found the conclusion, ask yourself why the author believes the conclusion to be true. The premises should provide the answer to the question. If you try the Why Test and the answer does not make sense, you have probably reversed the conclusion and premises. Let’s break down an example: Cream cheese contains 50 percent fewer calories per tablespoon than does butter or margarine. Therefore, a bagel with cream cheese is more healthful than is a bagel with butter on it. Use the Why Test to identify the conclusion and premises.

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First, find the conclusion. The word “Therefore” is a conclusion indicator. The conclusion of this argument is that “a bagel with cream cheese is more healthful than is a bagel with butter on it.” Next, let’s use the Why Test to confirm that we have correctly identified the conclusion. Ask, “Why is a bagel with cream cheese more healthful than a bagel with butter?” The answer is, “Cream cheese contains 50 percent fewer calories per tablespoon than does butter or margarine.” Since the information makes sense as support for the conclusion, we know we correctly broke down the argument. Had we reversed the premise and conclusion, the Why Test would have failed. It is not logical to conclude that, “Cream cheese contains 50 percent fewer calories per tablespoon than does butter or margarine” because “Therefore, a bagel with cream cheese is more healthful than a bagel with butter on it.” Always use the Why Test to separate the conclusion and premises.

Assignment 3

Drill: Conclusions and Premises Using indicator words and the Why Test, identify the conclusion and premise(s) in the following examples. 1. In a free society people have the right to take risks as long as they do not harm others as a result of taking the risks. Therefore, it should be each person’s decision whether or not to wear a seat belt. Conclusion:

3. Last year, the city of Melville increased the size of its police force by fifty officers. This year, there was a 10 percent decrease in the number of violent crimes reported in Melville. Clearly, a larger police force discourages criminal activity. Conclusion:

Premise(s): Premise(s):

2. Companies have found that giving workers the option of flexible hours leads to happier employees. Happier employees are more productive. Company X has flexible hours. Company Z operates on a strict 9 to 5 workday. Company X will certainly beat Company Z in worker productivity. Conclusion:

Premise(s):

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Find the Gap Now that you are getting the hang of reading for the conclusion and premises, let’s move on to the next step in analyzing an argument. You probably found the preceding argument unconvincing. That’s because, like most GMAT arguments, they contain faulty reasoning. Whether or not you agree with an argument’s conclusion, the reasoning leading to that conclusion likely contains a flaw. The test writers construct arguments with questionable reasoning so that they can test your ability to identify and describe these errors. Identify the flaws in an argument by looking for a gap between the conclusion and premises. The premises rarely provide enough evidence to lead convincingly to the conclusion. Find the gap by determining what is mentioned in the conclusion that was not mentioned in the premises. Once you have found the gap, you can make the argument work by filling in the assumption. The assumption is the unstated part of the argument that is required to connect the premises to the conclusion. Though not explicitly stated by the author, the assumption must be true for the argument to be well reasoned. If you want to weaken an argument, widen the gap between the conclusion and premises by attacking the assumption. Let’s examine the cream cheese argument again: Conclusion: A bagel with cream cheese is more healthful than is a bagel with butter on it.

Look for the gap between the premises and conclusion.

Premise: Cream cheese contains 50 percent fewer calories per tablespoon than does butter or margarine. The author broadened the scope of the argument from calories (in the premise) to the more general statement about health in the conclusion. Thus, there is a gap between “fewer calories” and “more healthful.” The argument’s assumption must link these two ideas. To fill in the gap, we must accept that having fewer calories per set amount is enough to qualify one food as more healthful than another. We also need to believe that people use similar quantities of butter, margarine, or cream cheese on a bagel, proving that the combination of bagel and cream cheese has fewer calories than the combination of bagel and butter. These two assumptions are necessary to make a valid argument. If we wanted to weaken the argument, we would need to attack the assumptions. For example, we could say that most people use significantly more cream cheese on their bagels than they do butter. Someone who uses more than twice as much cream cheese as butter would actually consume more calories. Even if calories were the only determining factor in healthfulness, the bagel with cream cheese would be less healthful. Think of GMAT arguments as an incomplete chain of reasoning: Premises

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Conclusion

Assignment 3

When you state the assumption, you provide the missing link that fills in the gap and completes the chain:

Premises

Assumption

Conclusion

Identifying the gap is the most important step in analyzing arguments. If you have difficulty finding the gap in an argument, slow down and think about the argument. If an argument seems logically sound, you have probably made the same assumption as did the author. Examine your thought process and think about why you believe the conclusion makes sense. Then think about whether any of your beliefs might be false. Ask yourself, “How could the premises remain true, but the conclusion be false?” If you can think of a way to invalidate the conclusion without disputing the premises, you have located the logical gap between the premises and conclusion.

COMMON FLAWS If you learn to recognize these common flaws, you will find it much easier to work arguments questions.

Causal Flaws Frequently, an argument’s premises state that two things happened, and the author concludes that one caused the other. Causal arguments are by far the most common type of logical flaw you will encounter on the GMAT. Look at an example: A study indicated that adults who listen to classical music regularly are less likely to have anxiety disorders. Clearly, classical music helps calm the nerves and lower anxiety.

What’s the gap between the conclusion and premises?

The author concludes that classical music helps calm the nerves and lower anxiety because the study found a correlation between listening to classical music and experiencing a lower likelihood of having anxiety disorders. However, the fact that two things are related does not prove that one caused the other. To make the causal link in this argument, we must assume that listening to classical music was the only factor responsible for lowering anxiety. We must rule out the possibility that any other factors played a role. We must also rule out the idea that having a lower anxiety level causes people to listen to classical music. To break the causal link, we could show that another factor explains the lower rate of anxiety disorders in people who listen to classical music. For example, if we knew that a majority of people who listen to classical music take anti-anxiety drugs, we might conclude that the drugs, not the music, lowered their anxiety. We could also weaken the argument by showing the causality is reversed. If we knew that calmer people were predisposed to enjoy classical music because it reflects their mood, the conclusion would be invalid. © Princeton Review Management, L. L. C.

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Sampling and Statistical Flaws When you encounter an argument based on percentages, numbers, or samples, the flaw usually relates to the failure of the data to prove the conclusion. If you look at all evidence skeptically, you will be able to find these arguments more easily. These types of arguments are less common than causal arguments, but they still appear on the GMAT fairly often. Sampling arguments reach a conclusion based on evidence about a subset of a group. They assume that the subset is typical and reflects the larger group. Arguments about survey results usually fall into this category. Look at an example: Contrary to popular belief, high school students overwhelmingly approve of the high school administrative staff. We know this to be true because the student council expressed admiration for the high school principal and her staff in the council’s editorial for the school paper.

What’s the gap between the conclusion and premise?

The author concludes that students approve of the school administration based on the student council’s opinion as expressed in the paper. The gap is between the student council and the general student body, and the author draws on a sample population to reach a conclusion about the whole population. To make the link, we must assume that the editorial is an accurate reflection of the feelings of the general student population. We could break the link by proving the student council’s view does not represent the views of the rest of the students. For example, maybe the student council is made up of sycophants who want to get favorable college recommendations from members of the administration. Whenever an author bases a conclusion about a general population on a sample or survey, remain skeptical. The author assumes that the part of the population sampled or surveyed is representative of the entire population. To strengthen a sampling argument, provide a reason why the sample is representative of the whole. To weaken a sampling argument, show that the sample is not necessarily representative of the whole. Statistical arguments hinge on a questionable interpretation of numerical data. Most often, the author confuses percentages with actual values. Look at an example: Ninety percent of the population of Prelandia lived in rural areas in 1800. Today, only 20 percent of the population lives in rural areas. Clearly, more people lived in the countryside two centuries ago.

What’s the gap between the conclusion and premises?

The author believes that since the percentage of people living in rural areas decreased, the actual number of people living in rural areas must have decreased. Arguments involving percentages are often math problems. Remember that percentages compare a part to a whole. In this case, the percentages tell us: Rural population Total population Whether or not that is true depends on how the past and present populations compare. To make the link, we need to prove that the total population in the past and present are comparable. For example, if the population was 100 people in 1800 and 100 people today, we could say that 90 people lived in rural areas in the past and 20 people live there today. The argument would be valid. To attack the argument, we need to prove that the total population has changed. For example if, the country’s population was 100 people in 1800, 90 of them lived in rural areas. If the population is 1,000 people today, then 20 percent would be 200 people. More people actually live in rural areas now, even though their percentage in the population has decreased. The argument is invalid.

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Assignment 3

Arguments that involve a comparison of percentages assume that the percentages are based on comparable totals. Be wary whenever an author uses information from percentages to draw conclusions about actual values and vice versa. To strengthen statistical arguments, prove that the author’s interpretation of the figures is valid. To attack statistical arguments, add information that calls the author’s interpretation of the statistics into question. Weakening statistical arguments most often involves demonstrating that the author incorrectly compared two percentages or confused percentages with actual values.

Analogy Flaws Some arguments use evidence about one thing to reach a conclusion about another. These arguments assume that two things are similar enough to sustain the comparison. Look at an example: Contrary to opponents’ charges that a single-payer health-care system cannot work in a democratic nation such as the United States, an overhaul of the American health-care system is necessary. Opponents of the single-payer system in the United States should remember that Canada, a nation with a strong democratic tradition, has run a viable single-payer health-care program for many years.

What’s the gap between the conclusion and premise?

The author concludes that because a single-payer health-care system works in Canada, it will work in the United States. The gap is between the United States and Canada, and we must assume they are similar enough to make the comparison valid. To strengthen this argument, we could add additional reasons why the two nations may be compared. We could weaken the argument by suggesting reasons why the comparison is not valid. For example, we could say that the differences in the populations and economies of the two nations mean that policies that work in one country won’t work in the other.

SUMMARY To analyze an argument: Find the conclusion and premises. Use the Why Test to ensure you have correctly identified each part. Identify the conclusion, and ask why it’s true. Restate the conclusion and premises in your own words, and write them down in shorthand form on your scratch paper. Weed out the useless information. Some arguments contain information that is neither the premise nor a conclusion. Sometimes the test writers provide background information to introduce a topic, or they use extra statements as a device to hide the information that you need. Using the Why Test will help you find the important information in the argument. Find the gap. Look for words, ideas, or conditions mentioned in the point that were not mentioned in the reasons. The weak point of every argument lies in the gap between the indisputable (for the purposes of the GMAT) reasons and the disputable point.

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Common Flaws Causal Arguments In a causal argument, the author assumes that an observed result can be explained by a single cause, factor, or reason. Causal assumptions can be phrased in a few different ways. An author might wrongly assume: • Because two things are related, one caused the other (correlation equals causality). • There are no other possible causes of or explanations for a result. • The causality did not occur in reverse. • To strengthen a causal argument, add a premise that supports the cause cited by the author or rule out other causes. To weaken a causal argument, add a premise that shows another cause of the result, that the cause and effect are reversed, or that one event can occur without the other. Sampling and Statistical Arguments When you encounter sampling or statistical arguments, accept the facts provided in the premises as true. Find the gap by questioning the conclusion the author draws using those facts. Strengthen these arguments by showing the sample or statistics are sufficient to prove the point. Weaken these arguments by showing the author misinterpreted the evidence. Show the sample is not representative of the whole or that the percentages are not representative of the total populations. Analogy Arguments When an author bases a conclusion on a comparison, he or she assumes that the items are similar and that what is true for one is true for the other. Strengthen these arguments by providing another reason why the items in question are comparable. Weaken these arguments by showing the items are not similar.

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Assignment 3

DRILL This drill gives you a chance to put together all you have learned so far about analyzing arguments. For each argument, find the point, reasons, and gap. Then try to either state the questionable assumption or what is wrong with the reasoning. Be on the lookout for the common flaws discussed above. 1.

A study of drinking habits shows that the rate of heart disease among those who drink one or two drinks a day (based on one drink = 1 ounce of 80-proof distilled spirits) is significantly lower than it is among those in the population at large. The study also shows that among those who drink excessively (six or more drinks each day), the rate of severe depression is much higher than it is among the general population. It was concluded from this evidence that level of alcohol consumption is a determining factor in the development of certain physical and psychological disorders.

Conclusion: Premise(s):

Conclusion:

Gap:

Premise(s):

Flaw/Assumption(s):

Gap: Flaw/Assumption(s): 2.

3. The ancient Egyptian pharaoh Akhenaten, who had a profound effect during his lifetime on Egyptian art and religion, was well loved and highly respected by his subjects. We know this from the fierce loyalty shown to him by his palace guards, as documented in reports written during Akhenaten’s reign.

Fortunately for the development of astronomy, observations of Mars were not very precise in Kepler’s time. If they had been Kepler might not have discovered that the curve described by that planet was an ellipse, and he would not have discovered the laws of planetary motion. There are those who complain that the science of economics is inexact, that economic theories neglect certain details. That is their merit. Theories in economics, like those in astronomy, must be allowed some imprecision. Conclusion:

4. Until he was dismissed amid great controversy, Hastings was considered one of the greatest intelligence agents of all time. It is clear that if his dismissal was justified, then Hastings was either incompetent or disloyal. Soon after the dismissal, however, it was shown that he had never been incompetent. Thus, one is forced to conclude that Hastings must have been disloyal. Conclusion: Premise(s): Gap: Flaw/Assumption(s):

Premise(s): Gap: Flaw/Assumption(s):

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5. Sixty adults were asked to keep a diary of their meals, including what they consumed, when, and in the company of how many people. It was found that at meals with which they drank alcoholic beverages, they consumed about 175 calories more from nonalcoholic sources than they did at meals with which they did not drink alcoholic beverages. Therefore, those wishing to restrict their caloric intake should refrain from drinking alcoholic beverages with their meals. Conclusion: Premise(s): Gap: Flaw/Assumption(s):

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Assignment 3

ANSWERS AND EXPLANATIONS Drill: Conclusions and Premises 1. Conclusion: It should be each person’s decision whether or not to wear a seat belt. Premise(s): In a free society, people have the right to take risks as long as they do not harm others as a result of taking the risks. 2. Conclusion: Company X will certainly beat Company Z in worker productivity. Premise(s): Flexible hours lead to happier employees, who will be more productive. Company X has flexible hours and Company Z does not. 3. Conclusion: A larger police force discourages criminal activity. Premise(s): Melville increased the size of its police force and saw a decrease in reports of violent crime.

Drill 1. Conclusion: Level of alcohol consumption is a determining factor in certain physical and psychological disorders. Premises: In the study, low alcohol consumption correlates with a low rate of heart disease, and high consumption correlates with a high rate of depression. Gap: The new idea in the conclusion is that alcohol was a “determining factor,” in other words, the cause of the observed results. This is a causal argument that views the correlation between alcohol and disease found by the study as evidence that alcohol is a causal factor in disease. To make that link, we must assume the only reason the first group had a low rate of heart disease was its low level of alcohol consumption. The only reason the second group had a high rate of depression was its high level of alcohol consumption. We also need to rule out a reverse of cause and effect. We must believe that disease had no effect on drinking habits. 2. Conclusion: Economic theory must be allowed imprecision. Premise: The Kepler example shows it is good that theories in astronomy were not always precise. Gap: The premise deals with astronomy, but the conclusion deals with economics. Flaw/ Assumption: The author quite clearly

compares astronomy and economics, yet she does not explain why it is acceptable to compare the two fields. We must assume that astronomy and economics are similar. This argument also has a causal aspect. The author assumes the Kepler example is sufficient to show that inexact theories have been beneficial to astronomy. She fails to consider that this might not be true in all cases. 3. Conclusion: The ancient Egyptian pharaoh Akhenaten...was well loved and highly respected by his subjects. Premise: According to reports written during Akhenaten’s reign, his palace guards were fiercely loyal to him. Gap: The conclusion talks about love and respect from his subjects, but the premise talks about loyalty of the guards. Flaw/Assumption(s): This argument has several flaws. The argument takes a particular fact (the reports of palace guards’ feelings toward the pharaoh) and interprets it to have some specific meaning. The argument assumes that the guards were representative of the pharaoh’s subjects as a whole. It also assumes that the reports accurately reflect the palace guards’ feelings and that loyalty is equivalent to love and respect. 4. Conclusion: Hastings must have been disloyal. Premise: If his dismissal was justified, he must have been either incompetent or disloyal. He was not incompetent. Gap: This one is a little tricky. The premise contains a condition (if his dismissal was justified), but the conclusion doesn’t take into account whether the dismissal really was justified. Flaw/ Assumption: The dismissal was justified. This is an example of a causal argument with a condition. When an argument involves a condition, you must assume the condition holds true. If we accept that the dismissal was justified, then only two possible causes remain, and if one cause is ruled out (incompetentence), the other must apply (disloyalty).

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5. Conclusion: Those wishing to restrict their caloric intake should refrain from drinking alcoholic beverages with their meals. Premise: According to their diaries, the sixty adults in a sample group consumed more calories from food when they drank with the meal than they did when they didn’t drink with the meal. Gap: The premise deals with a sample of sixty people, but the conclusion seems to be aimed more broadly. Also, the correlation between alcohol consumption and calories in the premise is interpreted as a causal relationship in the conclusion. Flaw/ Assumption: Again, as with all survey arguments, this one makes the assumption that the sixty adults in question were representative of those wishing to restrict their caloric intake (the group for whom a recommendation is made in the conclusion). This is, in addition, a causal argument. While it doesn’t seem especially plausible that the causation could be reversed—that, in other words, eating more could cause them to drink—this argument does definitely assume that there’s no third cause of both the drinking and the increased caloric intake at some meals. It may be, for instance, that the sample group (and people in general) most often drink when celebrating some special occasion over a meal, and that in such cases they also tend to eat more.

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Assignment 3

RATIOS AND STATISTICS This lesson focuses on ratios, mean, mode, median, range, and standard deviation.

RATIOS Earlier we dealt with fractions, decimals, and percents, which express part whole relationships. Ratios also express a relationship between two or more values, but with one big difference: ratios describe part relationships. part For example, consider a recipe for punch that calls for 2 parts grape juice to 3 parts orange juice. The relationship, or ratio, of grape juice to orange juice is 2:3. It doesn’t matter if you are mixing up one glass of punch or a whole pitcher— the relationship remains constant. Don’t confuse ratios with fractions. In this example, the 2 parts grape juice and 3 parts orange juice add up to 5 total parts. Thus, the fraction of grape juice in the mixture is 2 . 5 All of the following are equivalent ways of writing this ratio: • the ratio of grape juice to orange juice is 2 to 3 • the ratio of grape juice to orange juice is 2 : 3 • the ratio of grape juice to orange juice is 2 3 Knowing that the ratio of grape juice to orange juice is 2: 3 does not tell you the actual amount of grape juice or orange juice. We could have a small glass or a large vat of this punch, but if the recipe is followed, then dividing the amount 2 of grape juice by the amount of orange juice will always give . 3 A ratio tells you only the relative amounts of each quantity. It does not, by itself, tell you the actual amounts.

RATIO BOX If you’re given a ratio and want to determine actual amounts, you need more information. Let’s look at an example: 1. In a club with 35 members, the ratio of men to women is 2 to 3. How many men belong to the club? 2 5 7 14 21

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MEN Ratio Multiply by Actual Number

× =

WOMEN + =

× =

TOTAL = =

× =

We’ll use the Ratio Box to keep track of the information. Start by plugging in the ratio numbers from the problem, and add them up to get your ratio whole. You have one actual number, namely 35 members, which is the actual whole. Fill in that box, and figure out the multiplier. The ratio whole is 5 (2 + 3), and you must multiply 5 times 7 to get the actual whole, 35. To keep the ratio intact, the same multiplier must be applied to all parts. Thus, the actual number of men is 2 × 7 = 14. MEN Ratio Multiply by Actual Number

2 × 7 =

WOMEN + =

14

3 × 7 = 21

TOTAL = =

5 × 7 = 35

Let’s try ratios with data sufficiency. 2. Rachel throws a cocktail party for her friends. At the party, she serves martinis, screwdrivers, and boilermakers. How many martinis did Rachel serve at the party? (1) Rachel served martinis, screwdrivers, and boilermakers in a ratio of 5:7:9 respectively. (2) Rachel served a total of 35 screwdrivers. Statement (1) gives you a ratio, and nothing else. Remember that a ratio alone tells you nothing about the actual amounts. If you need proof, draw a ratio box and try to find the multiplier. You won’t be able to! Narrow it down to BCE. Now look at Statement (2), forgetting all about Statement (1). That Rachel served 35 screwdrivers does not tell us anything about the number of martinis or boilermakers she might have served. Eliminate (B). Now consider the statements together. Can you find a multiplier and fill in your ratio box? Since you know the ratio number for screwdrivers is 7, and the actual number of screwdrivers is 35, the multiplier must be 5 (35 = 5 × 7). The multiplier applies to all parts of the ratio, so you have enough information to determine the number of martinis served. However, don’t waste time solving a data sufficiency question. Once you realize that you CAN solve it, you know that the statements together are sufficient. The answer is (C), and you don’t need to fill in the whole box.

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Assignment 3

Here’s another type of ratio problem: 3. If 3u=5v, then the ratio of 5u to v is 1:3 3:1 4:3 15:1 25:3 Since the problem involves variables, let’s Plug In. What are some easy numbers to plug in for u and v? How about u = 5 and v = 3? In that case, the ratio of 5u to v would be 5 × 5 to 3, or 25: 3, so the best answer is (E).

Proportions Some problems set two ratios equal to one another. These fixed relationships are called proportions. For example, the relationship between hours and minutes is fixed, so we can set up a proportion:

1 hour 3 hours = . 60 minutes 180 minutes

The key to doing a proportion problem is to set one ratio equal to another, making sure to keep your units in the same places. Let’s try an example: 4. On a certain map, Washington, D.C., and Montreal are 4 inches apart. If Washington, D.C., and Montreal are actually 500 miles apart, and if the map is drawn to scale, then 1 inch represents how many miles on the map? 125 150 250 375 500 The information tells you that the ratio of inches to miles is 4: 500. Those are equivalent units, since 4 inches on the map is the same as 500 miles. Put those

4 inches . Next, set up the other side, 500 miles 4 inches 1 inch keeping units in the same places in the fraction: . Now solve = 500 miles x miles equal amounts in a fraction, like this:

for x. You can cross multiply to get 4x = 500 , so x = 125, which is choice (A).

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AVERAGES On the GMAT, average is also called arithmetic mean, or simply mean. All averages are based on this equation:

Average =

Total Number of Things

Drawing an Average Pie will help you organize your information.

Total # of Average things

Here’s how the Average Pie works. The total is the sum of the numbers you are averaging. The number of things is the number of quantities you are averaging, and the average is, of course, the average. Let’s apply the Average Pie to a simple example. Say you wanted to find the average of 3, 7, and 8. You would add up the numbers and then divide by 3  3 + 7 + 8 18  = = 6 . Here’s how to organize the same information in the average  3 3 

pie:

18 . .

. . 3

6

The horizontal line across the middle means divide. If you have the total and the number of things, divide to get the average. If you have the total and the average, divide to solve for the number of things. If you have the average and the number of things, multiply to get the total. As you will see, the key to most average questions is finding the total. You may also need more than one pie if a problem involves multiple averages. Draw a separate pie for every average in a problem. The Average Pie is a great tool because it: • Organizes the information clearly. • Allows you to solve for one piece of the pie when you have the other two. • Helps you to focus on what else you need to solve the problem, which is great for data sufficiency. 92

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Assignment 3

Let’s try a problem. 1. The average (arithmetic mean) weight of three people is 160 pounds. If one of these people weighs 200 pounds, what is the average weight, in pounds, of the two remaining people? 73

1 3

140

160 240 480 Draw an Average Pie, and fill in the first set of information. You know the average of three things is 160, so can you multiply to get the total (480). Notice that you can cross off (E) as a trap. The question asks for the average weight of the two remaining people, so you’re not done yet. Draw another Average Pie for the next average in the problem. There are two remaining people, but you don’t know their total weight. However, you can get their total weight by subtracting 200 from the total for all three (480). Since 480 – 200 = 280, you know the total weight of the other two people is 280. To get the average, just divide your new total by the number of people,

280 pounds = 140 . Choice (B) is the 2 people

answer.

RATES Rate problems are very similar to average problems. They often ask about average speed or distance traveled. Other rate problems ask about how fast someone works or how long it takes to complete a task. All of these problems involve this important relationship:

Rate =

Distance Amount of Work or Rate = Time Time

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Because this relationship is identical to that in the average formula, you can use the Rate Pie to organize your information:

Distance Time

Rate

Work Time

Rate

The Rate Pie has the same advantages as the Average Pie. Whenever you have two pieces of the pie, you can solve for the third. Let’s put the Rate Pie to work. 1. It takes Mike 1 hour and 30 minutes to commute from home to work at an average speed of 40 miles per hour. If Mike returns home along the same route at an average speed of 45 miles per hour, how long does the return trip take? 1 hour, 15 minutes 1 hour, 20 minutes 1 hour, 25 minutes 1 hour, 30 minutes 1 hour, 35 minutes Draw a Rate Pie, and fill in the information you know about the first trip: Mike’s rate was 40 mph, and the time it took was 1 hour and 30 minutes, or 1.5 hours. Now you can solve for total distance by multiplying: 40 × 1.5 = 60 miles. That’s the same distance he’ll need to travel on the way home, so make another Rate Pie for the return trip, and fill in that piece. Now you also know the return trip average speed is 45 miles per hour, so go ahead and put that in too. So the 60 4 1 time for the return trip can be found by = = 1 hours, or one hour and 45 3 3 twenty minutes. The answer is (B). Let’s try another one, this time with work. 2. A machine at the golf ball factory can produce 16 golf balls in 5 minutes. If several of these machines work independently and each machine performs at the same rate, how many machines are needed to produce 32 golf balls per minute? 3 6 8 10 13

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Assignment 3

Put the information you have into a Rate Pie. You know that one machine produces 16 balls in 5 minutes, so fill in those pieces. That gives you a rate of 16 = 3.2 balls per minute. Then set up a proportion, since each machine works 5 1 machine x machines = at the same rate, . Cross-multiply and you get 3.2 balls 32 balls 32 × 1 = 10 , choice (D). x= 3.2

MEDIAN, MODE, AND RANGE You’re already familiar with the mean, which is another word for average. Here are some other terms you need to know: Median is the number in the middle after your set of numbers has been arranged in ascending order. If the set has an even number of elements, the median is the average of the two numbers in the middle. Mode means the most frequently occurring number (or numbers) in the set. Range is the difference between the highest and the lowest numbers in the set. Okay then, in the following set of numbers, can you identify the mean, median, mode, and range? {2, 25, 10, 6, 13, 50, 6} Mean = Median = Mode = Range = To find the mean, add the numbers to find the total, and divide by the number of things in the set. The total is 2 + 25 + 10 + 6 + 13 + 50 + 6 = 112. There are 7 numbers in the set, so the average, or mean, is 112 ÷ 7 = 16. To find the median, put the numbers in order from least to greatest. That gives you 2, 6, 6, 10, 13, 25, 50. Because this set has an odd number of elements, the median is just the number in the middle, 10. If we take out the number 50 from this set, what is the new median? Now there is an even number of things, 6 + 10 = 8. so take the two numbers in the middle, 6 and 10, and average them: 2

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To find the mode, look for the number that occurs most often. Since only the number 6 shows up more than once, it’s the mode. To find the range, subtract the lowest number from the highest. That’s 50 – 2 = 48. Fortunately, the problems that involve these terms usually aren’t very difficult, as long as you don’t confuse their meanings.

STANDARD DEVIATION Standard deviation is another statistical term like mean, median, range, and mode. If you see a problem that uses the terms normal distribution or standard deviation, think about ETS’s favorite object in the whole wide world: the bell curve.

2% 2%

14%

34% 16%

34% 50%

14% 84%

2% 98%

Any normal distribution of a set of values can be plotted along this bell curve. A few key parts to our curve: • The mean is indicated by the line down the center of the curve. In questions dealing with standard deviation, the mean will either be given to you, or it will be easy to figure out.

Use the bell curve to organize your information.

• The standard deviation is a statistically derived specified distance from the mean. In the figure above, the standard deviations are represented by the solid lines. You will never have to calculate the standard deviation from the data alone. GMAT problems tell you the standard deviation for a set of data. • The percentages indicated in the picture represent the portion of the data that fall between each line. These percentages are valid for any question involving a normal distribution, and you should memorize them: 34: 14 : 2. The percentages correspond to the 1st, 2nd, and 3rd standard deviations on each side of the mean. Working with standard deviation on the GMAT is mostly a matter of drawing the curve and filling in the information. Let’s take the case of an exam scored on a scale of 200 to 800.

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Assignment 3

If the mean on this exam is 530, and the standard deviation (sd) is 110, what percent of test takers score between 420 and 750?

530

420

34%

34%

640

Use the bell curve to organize your information.

750 14%

14%

2%

- 3sd

2%

-2sd

-1sd

Mean

1sd

2sd

3sd

Start with the bell curve. Label what you know: the mean is 530, and each standard deviation is 110 points, so the first sd above the mean is at a score of 510 + 110 = 640, and the second sd above the mean is 640 + 110 = 750. You could go and figure out the third one, but the question only asks for scores up to 750, so you should save yourself the effort. You still need lower scores, so the first sd below the mean is 530 – 110 = 420. We have enough information to answer the question. Add up the percentages in your included portions of the curve, from 420 to 750, we have 34% + 34% + 14% = 82%. As long as you understand how to draw the chart and you memorize the percentages that correspond to each standard deviation, standard deviation problems are very manageable.

DATA SUFFICIENCY TRICKS AND TRAPS Some data sufficiency questions contain “tricks” designed to fool the unwary test taker. The trick usually involves some assumption that people tend to make or some esoteric math concept that people forget. Beware of the trap answers! Each of the following questions contains a trap. Try to spot and avoid the trap as you work the question. Don’t read the explanation until you’ve tried the question on your own. 1. What is the value of x? (1) x > 9 (2) x < 11 Statement (1) is not enough because x could be any number above 9, so BCE. Statement (2) is not enough because x could be any number below 11, so eliminate B. If you chose (C), you probably assumed that x was 10 because 10 is the only integer greater than 9 but less than 11. However, nothing in the question requires x to be an integer. For example, it could be 10.2 or 10.5. Since there’s no way to determine a single value for x, choose (E). © Princeton Review Management, L. L. C.

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2. If xy ≠ 0, what is the value of

( )

x 4 y 2 − xy

2

?

x 3 y2 (1) x = 2 (2) y = 8 The natural instinct is to go straight for (C) because it provides values for both variables. However, (C) is a trap. This question nicely illustrates why you should go through the step-by-step AD/BCE approach. If you substitute the information from Statement (1) into the expression, you get: 16 y 2 − 4 y 2 12 y 2 3 . Thus, Statement (1) alone is sufficient to determine = = 2 8y 2 8y 2 the value, and the only possible answers are (A) and (D). If you substitute the information from Statement (2) into the expression, you get: 64x 4 − 64x 2 64(x 2 − 1) (x 2 − 1) . Since you haven’t solved for a value, = = 64x 3 64x 3 x3 Statement (2) is insufficient, and the answer is (A). Another way to work this problem is to factor y2 out of each part of the expression right at the start. Once you realize that y2 cancels out of the expression, it’s clear that you only need to know the value for x. 3. John is driving from Town A to Town B. What is his average speed over the entire trip? (1) He drives the entire 120 miles in 3 hours. (2) His maximum speed during the trip was 50 miles per hour and his minimum speed was 30 miles per hour. Because you have distance and time, Statement (1) is sufficient to find the average speed, or rate (120/3 = 40 mph). The only possible answers are (A) and (D). If you chose (D), you probably assumed that Statement (2) also told you the average speed was 40 mph. What you learned from the first statement may have influenced your reading of the second statement. However, you can never average the averages; you must always derive the average from the total and number of things. Choose (A). 4. How far is Town A from Town C? (1) Town A is 160 kilometers from Town B. (2) Town B is 155 kilometers from Town C. Statement (1) isn’t enough because Town C is not mentioned, so choices (B), (C), and (E) remain. Statement (2) isn’t enough because it omits Town A. That leaves (C) and (E). If you chose (C), you probably assumed that towns A, B, and C are arranged in a straight line and that you could add up the distances (160 + 155 = 215). However, the actual order could be A—C—B, or the towns could form a triangle. Choose (E). 98

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Assignment 3

When you work data sufficiency questions be wary of: • The Integer Trap: Don’t assume numbers are integers unless the problem tells you so. • The Variables Trap: Don’t assume you need to find values for all the variables in a problem, especially if the question contains an expression with more than one variable. Try manipulating or simplifying the expression before you read the statements. • The Statements Trap: Forget about Statement (1) when you consider Statement (2). Be sure to consider each statement by itself. Combine the information in the two statements only if each statement alone is insufficient and you’re down to (C) or (E). • The Diagram Trap: Don’t assume a diagram looks a certain way unless the problem tells you so.

SUMMARY Ratios • A ratio expresses a part relationship. part • For any problem that gives you a ratio and an actual number, make a Ratio Box. • Plug In on ratio problems with variables.

Proportions • A proportion is an equation that sets two ratios equal to each other. • For problems that give you 3 out of 4 variables and asks you to solve for the fourth, set up your proportions (equal ratios), making sure to keep your units in the same places, and cross-multiply.

Averages • For average problems, use the equation:

Average =

Total Number of Things

• Use the Average Pie to organize your information. • For data sufficiency, ask yourself, “Do I have enough information to fill in the missing piece of the puzzle?”

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Rates • For problems dealing with rates of work or travel, use the equation: Rate = Distance or Rate = Amount of Work Time Time • Use the Rate Pie to organize your information.

Median, Mode, and Range • The mean is the average of all the numbers in a set. • The median is the middle number in a set. • The mode is the number (or numbers) that appear(s) most frequently in a set. • The range of a set is the difference between the highest number and the lowest number in the set.

Standard Deviation • On the GMAT, problems dealing with standard deviations, or normal distributions, are about the bell curve. • The mean, or average, is the center of the bell curve. • A standard deviation is a specific distance away from the mean. • The percentages 34%, 14%, and 2% correspond to the 1st, 2nd, and 3rd standard deviations on each side of the mean.

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Assignment 3

DRILL 1. If a certain concrete mixture contains gravel, water, and sand in a 1 to 3 to 5 ratio by weight, how many pounds of gravel would be needed to produce 72 pounds of the mixture? 72 40 24 9 8

5. The average (arithmetic mean) of eight numbers is 7. If two numbers are discarded, the average of 1 the remaining numbers is 6 . What is the 2 average of the two discarded numbers? 3 4

2. A disc jockey plays only hip-hop and countrywestern records. If the disc jockey plays four country-western records for every seven hip-hop records he plays, what fraction of the records he plays are country-western?

1

1 2

3

1 2

4 5

8

1 2

7 11 4 7 4 11 1 28 3. It’s rumored that in 1976, Elvis consumed three times his body weight in peanut butter and banana sandwiches. If Elvis’s body weight in 1976 was 250 pounds, and if a peanut butter and banana sandwich weighs four ounces, then how many such sandwiches did Elvis consume in 1976? (1 pound = 16 ounces) 750 1,000 1,500 3,000 4,000 4. What is the ratio of x to y, given that xy ≠ 0 ?

17 6. The average (arithmetic mean) of 6, 21, x, and y is 13, where x and y are integers with a product of 100. Which of the following could be x? 50 20 10 4 1 7. If a, b, c, d, and e are consecutive integers in increasing order, what is the average of the five numbers? (1) a = 13 (2) a + e = 30 8. For the set of positive, distinct integers {v, w, x, y, z}, the median is 10. What is the minimum value of v + w + x + y + z? 25 32 36 40 50

(1) 3x = 4y (2) x = 4

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9. The sum of the numbers in the set {5, 28, 10, 50, x, 7} is 120. What is the median of the set? 10 15 18 20 24 10. For the set of numbers {20, 14, 19, 12, 17, 20, 24}, let x equal the median, v equal the mean, w equal the mode, and y equal the range. Which of the following is true? v 1?  b (1) b – a = 9 (2) c > 1

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61 3 129 183

9. Is m2 an integer? (1) m2 is an integer. (2)

m is an integer.

Class Lesson 4

10.

(

50 5−2 − 2−2

)

4. Is m a multiple of 6? (1) More than 2 of the first 5 positive integer multiples of m are multiples of 3. (2) Fewer than 2 of the first 5 positive integer multiples of m are multiples of 12.

52 3 2 –

3 2

–

5. If 6 is a factor of a and 21 is a factor of b, is ab a multiple of 70? (1) a is a multiple of 4. (2) b is a multiple of 15.

21 50

6. Does s = t ? (1) s = t (2) s is both a factor and multiple of t.

21 50 –

21 25

11. If a is not equal to zero, is a–3 a number greater than 1? (1) 0 < a ≤ 2 (2) ab = a

FACTORS, DIVISIBILITY, AND MULTIPLES 1. What is the sum of positive integers x and y? (1) x2 + 2xy + y = 16 (2) x2 – y2 =8 2. Which of the following are roots of the equation

( )(

) = 0?

x x + 5 x2 − 4 x + 12

–2, 0, 5, –12 0, –5, 2, 12 –2, 0, 2, 5, –12 –5, –2, 0, 2 0, 4, 5 3. What is the remainder when integer n is divided by 10? (1) When n is divided by 110 the remainder is 75. (2) When n is divided by 100 the remainder is 25.

7. If n is an integer greater than 0, what is the remainder when 912n+3 is divided by 10? 0 1 2 7 9

FACTORIALS 1. If n is an integer greater than 5.3, then n! must be divisible by which of the following numbers? 7 11 12 13 14 2. If for all positive integers x and y, y > x, then is y!

even?

x! (1) y = 13 (2) y – x = 2 3. The fraction

11!

is equivalent to which of the

77

following? 273352 293452 283452 28345272 28345211

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4. Is a prime? (1) x! = a (2) x > 2 5. What is the value of xyz? (1) y! = 6 and x! > 720 (2) z is the least even integer greater than –1. 6. If x and n are positive integers, is n! + x divisible by x? (1) n > x (2) n is not a prime number.

PERMUTATIONS AND COMBINATIONS 1. Flippy’s Flowers is designing a special prom corsage that consists of one rose, one orchid, and one gardenia. If Flippy carries four types of roses, three types of orchids, and five types of gardenias, how many different corsages can Flippy design? 12 24 30 60 120 2. At a prestigious dog show, six dogs of different breeds are to be displayed on six adjacent podiums. If the Springer spaniel must be displayed on the leftmost podium, how many display arrangements of the six dogs are possible? 5 6 30 120 240 3. On Random Omelet Monday, a chef creates omelets by randomly choosing three out of a possible six fillings. How many different omelets can the chef possibly make? 9 18 20 120 720

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4. The co-op board of a certain residential building must consist of two men and three women. If there are six men and seven women who want to be on the committee, how many different makeups of the committee exist? 65 525 1,050 1,287 100,800 5. Eight Alaskan Huskies are split into pairs to pull one of four sleds in a race. How many different assignments of Huskies to sleds are possible? 32 64 420 1680 2520 6. In a group of 8 semifinalists, all but 2 will advance to the final round. If in the final round only the top 3 will be awarded medals, then how many groups of medal winners are possible? 20 56 120 560 720 7. Alan has a flock of sheep from which he will choose 4 to take with him to the livestock show in Houston. If Alan has 15 distinct possible groups of sheep he could take to the show, then which of the following is the number of sheep in his flock? 30 15 7 6 5

Class Lesson 4

ANSWERS AND EXPLANATIONS Roots and Exponents 1. A

2. A

If you are asked a specific question to which you are given specific numerical answer choices, you should always Plug In the Answers. Though we typically start with answer choice (C), here we want to start with the choice that will be easiest to calculate, which is (B). If we plug in choice (B), we get the answer 124 = –126. Clearly this answer is too large, so we need to try something smaller. Try choice (A) since it’s the only smaller choice. It works.

6. A

the intention cannot be to get you to just do the math; to find a common denominator here would require ugly calculations. So, look for a simpler way to solve the question. The presence of the exponents and all the multiples of two

Statement (1) tells us that a = 0, since only division by 0 is undefined. Since b is distinct from a, it cannot be 0, and this statement is sufficient to determine that ab is 0. Eliminate BCE and keep AD. Statement (2) tells us nothing about a, and so is insufficient; eliminate choice (D).

should give you a hint that expressing things as powers of two might be a good way to get a handle on the problem. If we express the

3. D Plug In possible values for n. In both statements, you cannot plug in a value for n that is greater than 3 without contradicting the statement, so each statement is sufficient, and the answer is (D).

numerators as powers of two, then we would get

Statement (1) tells us that

1 1 1 1 + 12 + 12 + 12 . Now adding our fractions 12 2 2 2 2 4 is really easy, and we get 12 . Unfortunately, 2

a is a fraction less than b

they aren’t done making us work. Since our

one, and any fraction less than one, when raised

answer doesn’t show up in the answer choices,

to a positive power, will remain a fraction less

we have to reduce again:

than one. Statement (1) is sufficient. Eliminate BCE. Statement (2) does not tell us anything about a a ; if is an integer greater than one, then b b c  a a is a  b  will also be greater than one, but if b c  a fraction less than one, then   will not be  b

greater than one. Statement (2) is insufficient;

1 2 2 2 23 . Next, reduce each of the + + + 212 213 214 215

fractions that can be reduced, and you have

4. C Convert the speed that’s in scientific notation into a “regular” number to avoid confusion. 3.316 × 102 = 331.6. 1,500 is about 5 times as big, so the answer is (C). 5. A

The first thing you should recognize here is that

7. A

4 22 1 = = 10 . 12 12 2 2 2

This question begs to be translated a little bit 1 before even looking at the statements. x − y = y . x 1 The only way y can be negative is if x is negative x and y is odd. Statement (1) tells us that x is not negative, and since it is not negative, there is no way for x − y to be negative. Eliminate BCE. Statement (2) doesn’t tell us whether y is even or

eliminate (D).

odd so that doesn’t help answer the question.

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8. B

1 ). You should now have the inside of the 52  1 1 parentheses as  –  . Now distribute the 50  25 4  25 and reduce, and you get 2 – . The next step is 2

This question calls for straight math, covering fractions and exponents. The easiest way to start is to re-express 48 as powers of 2 and 3 and then distribute. Since 48 = 3 × 16 , we can write 48 as

to deal with the denominator. Dividing by 52 is

3 × 24. Then we can multiply that by each of the

the same as multiplying by

1 . We can then 52 25 25 distribute, which gives us . The last – 2 2 × 25

fractions in the parentheses and then reduce: 3 × 24 3 × 24 24 3 × 24 , , and = 3 = = 3 × 2 2 = 12 . 3 24 32 22

step is to reduce and then subtract the fractions. 2 1 4 − 25 21 = − . Don’t forget to bowtie to − = 25 2 25 × 2 50

Now add these all together: 3+

24 16 61 + 12 = 15 + = . Now, you’re dividing 3 3 3

the whole thing by 3 2 , which is the same as multipling by 9. B

10. C

1 1 61 1 61 or . . × = 2 9 3 9 27 3

Start by translating the question and understanding the pieces of the puzzle given and the pieces needed. To answer this question, we need to know whether m is an integer. Statement (1) is insufficient because m could equal 2 or m could equal 2 . Eliminate AD. Statement (2) tells us that m must be an integer because it must be the perfect square of an integer, and any integer squared is also an integer. This question tests basic math in a somewhat complex manner; it combines exponent, fraction, and distribution rules. First, we should probably re-express the numbers with negative exponents as fractions in order to multiply them by 50. Remember, a negative exponent is really just 1 over a positive exponent (e.g. 5–2 is the same as

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make the subtraction easier. 11. E

This question begs for a little translation and 1 simplification; it is another way to say 3 . For a 1 to be greater than 1, a must be a positive a3 fraction less than 1. Statement (1) does not resolve whether a is a fraction or not. Eliminate AD. Statement (2) only tells you that b is 1; it tells us nothing about a. Eliminate (B). When we look at the statements together, we know nothing more about a than we knew in Statement (1), so together they are still not sufficient.

Factors, Divisibility, and Multiples 1 D Start with Statement (1). If we factor the equation given, it yields (x + y)2 = 16, so x + y = 4 (note that we’re told that x and y are positive), so Statement (1) is sufficient. Eliminate BCE. Statement (2) can also be factored, and yields (x + y)(x – y) = 8. This tells us that x + y and x – y must be factors of 8. Eight only has four factors, 1, 2, 4, 8. If we consider each possible factor in turn, and if x and y are positive integers and must equal one of these factors, there is no way that x + y can equal 1. If x + y must equal 2, then x and y must both be 1, but in that instance x – y would not equal 4, thus x + y cannot be 1. If we continue to try each factor, the only factor of 8 that x + y could be is 4, thus this statement is also sufficient.

Class Lesson 4

anything to change the units digit, and when you divide a number by 10, its remainder will always be its units digit. No matter what value you plug in for n, we’re going to be raising 9 to an odd power, so the units digit and the remainder will both be 9.

2. D The roots of an equation are those values that make the equation equal 0. So all we have to do is find what values will make the equation equal 0. If you don’t see anything that will make the equation equal 0, then just plug in the answers. If you do, then look for your numbers in the answer choices, and eliminate anything without your numbers. The roots are –5, –2, 0, and 2. 3. D Look at Statement (1). Any multiple of 110 will have a units digit of 0. Add 75, and you’ll have a units digit of 5. The remainder of any integer divided by 10 is its units digit, so Statement (1) is sufficient. The same process will reveal that Statement (2) is sufficient. The answer is (D). 4. B

Start with Statement (1). Multiples of 6 (6, 12, 18, 24, and 30) would yield an answer of “yes.” Multiples of 3 (3, 6, 9, 12, 15) would yield a “no”. Thus Statement (1) is insufficient. Eliminate AD. Approach Statement (2) the same way. The information we are given in this statement doesn’t allow us to use 6 or any multiple of 6 for m, thus answering the question with a definitive “no!”.

5. B This question is all about factoring. We need to determine whether 70 is a factor of ab, and the easiest way to do that is to break 70 down into its prime factors. 7 × 5 × 2 = 70. So if ab is divisible by 7, 5, and 2, then it’s divisible by 70. The question itself lets us know that 70 is divisible by 2 (since 6 is a factor of a) and by 7 (since 21 is a factor of b), so all we need is proof that it is divisible by 5. Statement (1) does nothing to help, but Statement (2) shows that b is divisible by 5, and so is sufficient. 6. B

7. E

Statement (1) is insufficient because s and t could both be 1, which would be equal, or s could be 4 and t could be 2. Eliminate AD. What we are given in Statement (2) answers the question because the only number that can be both a factor and a multiple of t is t, thus s must be equal to t. When something looks like an insane amount of work, start looking for a shortcut. In this case, the shortcut is a pattern: 91 = 9. 92 = 81. 93 = 81 × 9 = 729. Multiply that by another 9? You’ll get a number ending in 1. Then one ending in 9. And so forth and so on. So the bottom line is that whenever 9 is raised to an odd power, the units digit is 9. When it’s raised to an even power, the units digit is 1. Adding 10 to a number won’t do

Factorials 1. C

If n is greater than 5.3, then the smallest n! can be is 6!. Since 6! = 6 × 5 × 4 × 3 × 2 × 1, it is definitely divisible by 12, because any n! bigger than 6 will include both a 6 and a 2, thus making it a multiple of 12. Also, n! does not have to be divisible by anything greater than 6, so 7, 11, and 13 are eliminated as are any multiples of those numbers, like 14.

2. B

Statement (1) only tells us what y is. Without knowing something about x, we cannot find out if

y! 13! is even or odd, because if x is 11, then x! 11!

would give us 13 × 2, which is an even number, while if x were 12,

13! would give us 13, which 12!

is an odd number. Eliminate (A) and (D). Statement (2) tells us the relationship between x and y by telling us that they are 2 apart. We now know that no matter what y is, x will be two less. So when you divide y! by x!, you will always be left with the two highest numbers, one even and one odd, and an even times an odd will always be even.

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3. C

4. C

5. B

6. A

Write out the factorial and then cancel everything that you can. Since 11! includes both a 7 and an 11 we can cancel those numbers with the 77 in the denominator leaving us with 10 × 9 × 8 × 6 × 5 × 4 × 3 × 2 and making answers (D) and (E) incorrect. Next, a glance at the answers shows that we don’t need to solve the equation, but rather just put it in an exponential form, so the next step is to express the remaining numbers as products of 2’s, 3’s, and 5’s. 10 can be expressed as 2 × 5, 9 as 3 × 3, and so on. Your final step would be to apply your exponent rule that tells you when you multiply exponents with the same base, you add the exponents. Statement (1) does not tell us anything about the value of x or a. We can’t say whether it’s prime or not, because if x is 2, then a is prime, but if x is anything other than 2, then a is not prime. Statement (1) alone is insufficient. Eliminate AD and keep BCE. Statement (2) says nothing about a, thus Statement (2) is insufficient alone. Eliminate B and keep CE. Taken together, we know that x is greater than 2, and so a is the product of at least 3 integers (3!). Since a prime number has only 2 factors, a cannot be prime, and the correct answer is (C). Statement (1) tells us what the value of y is, but does not give us the exact value of x or tell us anything about the value of z, and so is insufficient. Eliminate AD. Statement (2) tells us that z is 0, and thus we don’t need to know anything about the value of any of the other variables. This one is tough. To understand the relevance of Statement (1), you have to recognize the following: • A factorial is divisible by all positive integers less than or equal to the integer you are taking a factorial of. For example, x! is divisible by all positive integers smaller than x. • If b is a multiple of y, then if you add y to b, the result will still be divisible by y. For example, 12 is divisible by 3. If you add 12 + 3 it will still be divisible by 3. Alternatively, plug in values for x and n and you will find out the facts mentioned above, but that’s a lot of messy work. Statement (1) is sufficient. Keep AD, eliminate BCE. Statement

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(2) tells us nothing about x nor its relationship to n. Stating that n is NOT prime means it could be a vast number of values. Thus Statement (2) is not sufficient.

Permutations and Combinations 1. D If Flippy chooses one of each flower, he has 4 × 3 × 5, or 60, different corsages. 2. D The Springer spaniel must be on the far left. After that, 5 dogs could be on the second podium, 4 on the third, and so forth. The total number of arrangements is 1 × 5 × 4 × 3 × 2 × 1, which equals 120. 3. C

This is a combination problem. To choose 3 out of 6 fillings, the formula is

4. B

6 × 5 × 4 120 = = 20 . 3 × 2 ×1 6

First, find the number of ways you can choose 3  7×6×5 of 7 women:  = 35 and the number of   3×2×1 6×5  ways you can choose 2 of 6 men:  = 15 .  2×1  Multiply the numbers together: 35 × 15 = 525.

5. E

Here we have another combination problem; you must find out how many ways you can create the four teams. For the first team, you 8×7 = 28 possibilities. For the second 2×1 6×5 team, you have = 15 possibilities (you 2×1 have

have only 6 options because 2 dogs were assigned to the first sled). For the third sled, you 4×3 = 6 possible groups. For the final 2×1 2×1 team, you have = 1 group. To arrive at your 2×1 have

final answer, just multiply the numbers together.

Class Lesson 4

6. B

The entire discussion of rounds is a red herring. The question is asking for possible combinations of the final three, and it is possible for any of the original 8 contestants to have advanced to the final round, thus we need to pick 3 out of 8, and 8×7×6 order doesn’t matter. = 56 . 3×2×1

7. D This is a combination question that is somewhat tougher because we aren’t given the number of sheep that were initially in the flock, or so it seems. Plug In the Answers! In the answers we are given choices for the number of sheep in the flock; all we have to do is try them out until one gives us the right number of distinct groups when choosing 4 out of the flock. Start with choice (C). If there are 7 sheep in the flock, then we need to find out how many ways we can choose 4 of 7 sheep when order does not matter. 7 × 6 × 5× 4 . Since The math would look like: 4 × 3× 2 × 1 this yields 35, there must be fewer than 7 sheep. Try a smaller number. Six works.

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CRITICAL REASONING 2 There are a few types of critical reasoning questions that do not require you to identify the conclusion, premise, and gap. It is particularly important to identify these question types before you begin working the question.

INFERENCE QUESTIONS Inference questions ask you to infer or conclude something based on the passage. In other words, you have to assume that the information in the passage is true and find the answer that is a true statement. You will rely heavily on POE when you work inference questions. For each answer choice, ask yourself, “Must this be true?” If the answer is no, eliminate that choice. Let’s review what you learned about inference questions in the pre-class assignment. Step 1: Read and identify the question. Inference questions typically ask: • Which of the following can be inferred from the information above? • Which of the following conclusions is best supported by the passage? • Which of the following conclusions could most properly be drawn from the information above? • Which of the following must be true on the basis of the statements above? Step 2: Work the argument. Read the passage. Don’t look for the conclusion and premises. Step 3: Predict what the answer should do. Most of the time, you will not be able to come up with an answer in your own words. Simply keep in mind that you want the answer best supported by the facts.

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Step 4: Use POE to find the answer. Look for answer that must be true. Avoid answers that: • Go Beyond the Information Given (B.I.G). Eliminate choices that bring in new information, require outside knowledge, or need additional assumptions. • Go against information in the passage. • Are broader or more extreme than the passage. Correct inference answers frequently use words such as may, might, or sometimes. Avoid answers that include strong words such as all, must, or never, unless they are clearly supported by the argument. • Could be true but cannot be proven using the facts in the passage. 1. According to a recent study, fifteen corporations in the United States that follow a credo of social responsibility are also very profitable. Because of their credos, these fifteen corporations give generously to charity, follow stringent environmental protection policies, and have vigorous affirmative action programs. Which of the following can be correctly inferred from the statements above? Following a credo of social responsibility helps to make a corporation very profitable. It is possible for a corporation that follows a credo of social responsibility to be very profitable. A corporation that gives generously to charity must be doing so because of its credo of social responsibility. Corporations that are very profitable tend to give generously to charity. Corporations that have vigorous affirmative action programs also tend to follow stringent environmental protection policies.

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Class Lesson 5

2. The two divisions of a high-tech company have performed quite consistently over the past five years. In each year, the telecommunications equipment division accounted for 35 percent of profits and 15 percent of revenues, and the high-speed internet division made up the balance. Which of the following can properly be inferred regarding the past five years from the information above? The telecommunications equipment division has made higher profits per dollar than the high-speed internet division. Sales for both divisions have remained flat over the five years. The high-speed-internet market involved tougher competition than the telecommunications equipment market during the past five years. Management devoted a greater number of company resources to the telecommunications equipment division than to the high-speed internet division over the past five years. More profitable products made up a higher percentage of the products offered by the telecommunications division. 3. A combination of anxiety and external pressure leads to nausea. All the auditioners for the new reality show Wanna Be’s suffer from external pressure. Some of the auditioners feel anxiety about performing well for the producers, but others do not feel anxious. The producers of Wanna Be’s like to choose contestants who feel anxiety. Which of the following conclusions is most strongly supported by the passage above? The auditions for Wanna Be’s cause more performers to feel external pressure than do auditions for other shows. Most of the people who audition become contestants on Wanna Be’s. There is more nausea among auditioners for Wanna Be’s than among producers of Wanna Be’s. No auditioner who doesn’t feel anxiety has nausea. Most of the auditioners for Wanna Be’s who become contestants have nausea.

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4. In broad thermodynamic terms, the distinction between solar energy and energy derived from fossil fuel is artificial. Fossil fuel molecules represent the decayed remains of plants. All of the energy these fossil fuels contain once resided in the sun and was, so to speak, trapped by plants here on Earth through the process of photosynthesis, whereupon it was housed, principally, within the carbohydrate molecules of which the plants were composed. The process of burning unleashes that energy, and when we run our lights, factories, and automobiles by burning fossil fuels _______________. Which of the following is the most logical completion of the passage above? we deplete our stores of an ever more precious resource we use energy that is, in fact, derived from the sun we spend the legacy left to us by our prehistoric ancestors we mimic the process of energy generation that exists in the sun we return to the sun that which originally resided there

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Class Lesson 5

RESOLVE/EXPLAIN QUESTIONS Some questions ask you to resolve an apparent paradox or explain a discrepancy. On the GMAT, a paradox is a seemingly contradictory pair of facts that is explained by one of the answer choices. Let’s review what you learned about resolve/explain questions in the preclass assignment: Step 1: Identify the question. Look for key words like resolve and explain. Step 2: Work the argument. Read the passage. Identify the facts in conflict. Look for words like but, yet, and however to find the paradox. Step 3: Predict what the answer should do. Phrase the question that the correct choice will answer. Ask, “Why X but also Y”? Step 4: Use POE to find the answer. The correct answer will provide additional information that allows both facts to be true and clears up the paradox. Avoid answers that: • Do nothing to clear up the conflict. • Make the conflict worse. • Address only one side of the conflict. 1. In September of last year, the number of people attending movies in theaters dropped precipitously. During the next few weeks after this initial drop the number of filmgoers remained well below what had been the weekly average for the preceding year. However, the total number of filmgoers for the entire year was not appreciably different from the preceding year’s volume. Which of the following, if true, resolves the apparent contradiction presented in the passage above? People under the age of 25 usually attend films in groups, rather than singly. The gross income from box office receipts remained about the same as it had been the preceding year. For some portion of last year, the number of people attending movies in theaters was higher than it had been during the previous year. The number of people attending movies in theaters rises and falls in predictable cycles. The quality of films released in September and October of last year was particularly poor.

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2. Although the mathematical validity of the laws of probability is indisputable, most people do not trust the dictates of these laws. Even among people who claim to have studied probability theory, for instance, a majority express a greater fear of flying on commercial airlines than of driving an automobile on our nation’s highways, despite the fact that the probability that one would suffer an automobile-related death or injury by choosing to drive is more than twenty times the probability of an airline-related death or injury if one chooses to fly. Which one of the following, if true, provides the best explanation for people’s mistrust of the laws of probability in the case described above? A complete understanding of the laws of probability requires a thorough knowledge of advanced statistical analysis techniques. People who studied probability theory in an academic environment may be ill equipped to apply that knowledge to real-world situations. People tend to suspend their belief in probability when they feel somewhat in control of their own fate. The probability of automobile-related injury or death is not significant enough to dissuade many people from driving. The greatest risk to the individual driver in terms of automobile-related injuries or fatalities are the actions of the other drivers on the road.

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Class Lesson 5

3. In France, the nuclear-generated electricity potential—the approximate peak output ability of France’s nuclear power plant network—is twice what it was fifteen years ago. Yet during this period no new nuclear power plants were built in France; in fact, five plants were decommissioned, due to various spending and safety concerns. Which one of the following, if true, best explains the discrepancy described above? Over the last fifteen years, the demand for electricity in France has outstripped the supply available from domestic sources. The current rate of increase in domestic demand for electricity in France is not as great as it was fifteen years ago. The planned expansion of the nuclear power plant network in France has been held up due to increased concerns over the safety of such facilities. The price of domestically generated electricity in France has declined significantly over the last fifteen years. The average French power plant has the potential to generate more electricity than it did earlier, due to advances in nuclear powergeneration technology over the last fifteen years.

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MINOR QUESTION TYPES A few other question types occasionally appear on the GMAT. To answer these questions, you’ll need to break down the arguments and understand their structures.

Evaluate-the-Argument Questions These questions deal with the assumptions in an argument. Here’s an example of an evaluate-the-argument question: 1. Ergonomically designed computer keyboards tend to lose their “play”—the responsiveness of the keys— more quickly than do traditional keyboards. A software designer has suggested that it is in fact the curvature of the key rows and not increased typing speed that is to blame. Due to the bent shape of the board, it is more difficult for the average user to clean between the keys, resulting in a gradual deadening of the spring mechanisms. The answer to which of the following questions will most likely yield significant information that would help to evaluate the software designer’s hypothesis? Do traditional keyboards and ergonomically designed keyboards utilize the same plastics? Does sprinkling a keyboard with dust impede the spring action beneath the keys? Does a keyboard with deadened play make typing more difficult? Do computer manufacturers receive more complaints about ergonomically designed keyboards than about traditional keyboards? Are software designers more likely than other users to utilize their keyboards when working?

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Here are the steps for working evaluate-the-argument questions: Step 1: Read and identify the question. Evaluate-the-argument questions often use words such as evaluate or assess. Also, the answer choices are often phrased as questions. Step 2: Work the argument. Identify the conclusion, premises, and gap. Step 3: Predict what the answer should do. Think about what you would need to know to fill in the gap. The correct answer will deal with one of the assumptions. Step 4: Use POE to find the answer. The format of the answer choices can be confusing. To alleviate this problem, try turning the questions into answers. The correct answer will either weaken or strengthen the argument. Eliminate the obviously wrong answers and choose the best answer from the ones that are left. 2. The recent surge in fear over the virulence of the Ebola virus is irrational and unfounded. While in 1996 only 66 deaths worldwide were directly attributed to Ebola, some 603 deaths were caused by the influenza virus in the United States alone in that same year. Yet no such hysteria has surrounded influenza, despite the significantly higher number of fatalities. Which of the following pieces of information would be most useful in evaluating the logic of the argument presented above? The geographical distribution of deaths directly attributable to the Ebola virus The incubation periods for a range of tropical viral diseases The probable cause of an outbreak of infection by the influenza virus in selected regions of the United States The relative survival rates for individuals infected with the Ebola virus and for those infected with the influenza virus The numbers of deaths attributable to other, non-viral diseases with similar origins

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Parallel-the-Reasoning Questions A parallel-the-reasoning question is really six questions in one. You not only need to identify the reasoning in the argument, but you must also identify the reasoning in each of the five answer choices. Fortunately, these questions rarely appear on the GMAT. Here’s an example: 1. If we reduce the salaries of our employees, then profits will increase by 35 percent. Because we must increase our profits, it is clear that employee salaries must be reduced. Which of the following most closely parallels the reasoning used in the argument above? If I eat less food, I will lose weight. Since I started skipping breakfast, I have lost ten pounds. If I work four more hours each week, I will earn enough money to afford a new hobby. Because I would like a new hobby, I will collect coins. If God does not exist, then there is no basis for morality. Because some actions are morally wrong, God must exist. If there is an economic recession, then salaries will be reduced. Because salaries are not decreasing, there is not an economic recession. If there were more commercials, the number of television watchers would decline. Because the number of television watchers should be reduced, the number of commercials should be increased.

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Class Lesson 5

MATH 5 Geometry on the GMAT is mostly a matter of knowing some facts and formulas and learning how to apply them. Even the toughest geometry problems test fairly basic ideas, but they layer a lot of them together and combine them in unusual ways.

RIGHT TRIANGLES As you learned, a right triangle is a triangle that contains a 90° angle. Right triangles obey the same rules that all triangles obey, but they also have some special properties. The relationship among the sides of any right triangle is expressed by the Pythagorean Theorem. If you know two of the sides of any right triangle, you can always find the third with the Pythagorean Theorem.

If a and b are the lengths of the legs, and c is the length of the hypotenuse, then:

The Pythagorean Theorem applies only to right triangles.

a2 + b2 = c2

c a

b

Common Right Triangles Certain right triangles show up quite frequently in GMAT problems. Any set of three numbers that satisfy the Pythagorean Theorem—and can therefore be the lengths of the sides of a right triangle—is called a Pythagorean triple. So, 3: 4: 5, 6: 8: 10, and 5: 12: 13 are Pythagorean triples.

5 3

4

10 6

8

13 5

12

Memorize these triangles and keep an eye out for them on the test. Recognizing them will save you time and effort. Notice that the 6: 8: 10 triangle is just a multiple of the 3: 4: 5. Pythagorean triples are ratios, so other multiples of these triangles will work too.

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B 4 C

A

D

12

1. In the figure above, if the area of triangle ABD is 30, what is the sum of AB and BC? 5 8 12 16 19 A

B

C

2. What is the area of triangle ABC? (1) AB = 3 and AC = 5 (2) B is a right angle. There are two other common right triangles you should know: • The 45°: 45°: 90° right triangle. The ratio of the sides is a: a: a 2 . • The 30°: 60°: 90° right triangle. The ratio of the sides is a: a 3 : 2a. Memorize the relationships among the sides:

45

a 2

a

a

45

A 45°: 45°: 90° triangle is also known as an isosceles right triangle. If you split a square diagonally, you create two 45°: 45°: 90° triangles.

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Class Lesson 5

30 2a a

3

60 a

Splitting an equilateral triangle in half creates two 30°: 60°: 90° triangles. To apply the Pythagorean Theorem, you need to know the lengths of two sides. If you know the angle measures of a 45°: 45°: 90° or 30°: 60°: 90° triangle, you only need to know one side to figure everything else out. Sometimes it’s helpful to use the approximate values of answers: •

2 ≈ 1.4

•

3 ≈ 1.7

2 and

3 to estimate

Hint to remember the ratios: The 45°: 45°: 90° triangle has 2 distinct angles, so it goes with √2 . The 30°: 60°: 90° triangle has 3 distinct angles, so it goes with √3 .

Y

X

Z

3. Triangle XYZ in the figure above is an equilateral triangle. If the perimeter of the triangle is 12, what is its area? 4 4 3 8 12 8 3

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B

C

6

A

D

4. In the figure above, what is the area of square ABCD? 4 10 12 18 24

COORDINATE GEOMETRY Many coordinate geometry problems are really about right triangles. If you need to determine the length of a line that’s not parallel to the x-axis or the y-axis, turn it into a right triangle problem.

y

R(3, 10)

S(9, 2)

1. In the coordinate grid above, what is the distance between point R and point S? 10 6 3 12 9 2 20

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Class Lesson 5

2. In a coordinate grid, if the points D(–1, –1), E(–1, 1), F(a, 1), and G(a, –1) are the vertices of a rectangle with a diagonal length of 2 5 , then what is the value of a? 3 5 3 5 9 7 3

Slope Slope is the measure of the steepness of a line. The steeper the line, the greater the absolute value of the slope. Slope is calculated by taking the vertical change between any two points on the line, and dividing it by the horizontal change between those two points. This is often called putting the rise over the run.

Slope = Rise = Vertical Change = y 2 − y1 Horizontal Change Run x 2 − x1 where (x1, y1) and (x2, y2) are points on the line It doesn’t matter which coordinate you call y2 or y1 as long as you use the corresponding x2 and x1.

3. What is the slope of the line that passes through the points (1, 4) and (5, 2)? –2 –

1 2

1 2 2 3 2

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Line Equations For some problems you need to know the general equation of a line. All the points on any non-vertical line must fit the line equation, which is usually expressed in the following form:

y = mx + b x, y = variables that stand for the coordinates of any point on the line m = slope of the line b = y-intercept = y-coordinate of the point (0, b) where the line crosses the y-axis For example, the line with the equation y = above the origin and has slope

3 2 1

4 x + 5 crosses the y-axis 5 units 3

4 . 3

1 2 3 4 5 6

-1 -2 -3

4. In the coordinate system shown above, the shaded region is bounded by straight lines. Which of the following is an equation of one of the boundary lines? y=1 y=2 x=0 2 y= y=

3 3 2

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x–2 x–2

Class Lesson 5

5. What is the x-intercept of the line defined by the equation y = 4x + 5? (5, –4) (4,

5

)

At the point where a line crosses the y-axis, the xcoordinate is 0. At the point where a line crosses the x-axis, the y-coordinate is 0.

4 (–

5

, 0)

4 4 ( , 0) 5 (0, –

5

)

4

SOLID GEOMETRY Let’s look at a few ways the GMAT might test you on three-dimensional figures. 1. What is the greatest distance (in inches) between any two corners of a rectangular box with dimensions of 6 inches, 8 inches, and 10 inches? 10 inches 12 inches 10 2 inches 10 3 inches 24 inches

The diagonal (i.e., the longest distance between any two corners of a rectangular solid) can be found using the following formula: a2 + b2 + c2 = d2 where a, b, and c are the dimensions (length, width, and height) of the figure.

D

C A

B

Remember that figures may not be drawn to scale in data sufficiency problems.

2. In the rectangular solid above, if AB = 5, what is the surface area of the solid? (1) BC = 8 (2) The volume of the solid is 80. © Princeton Review Management, L. L. C.

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x

3. A rectangular label of width x has been wrapped around the cylinder above, encircling the cylinder without overlap. If the radius of the cylinder is 6, and the label has the same area as the base of the cylinder, then what is the value of x? 3 5 6 6π 9π

OVERLAPPING FIGURES Some geometry problems involve multiple geometric shapes that overlap or are inscribed in one another. The question typically tells you something about one of the shapes and then asks you something about the other. The key to these problems is to figure out what the two shapes have in common. B

C

A

D

1. Rectangle ABCD has length 8 and width 6. What is the area of the circle? 10π 25π 50π 64π 100π

M

y

N

O

x L

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2. If the area of the circle above with center O is 64π, what is the area of triangle LMN? (1) x = 2y (2) OL = LM

Class Lesson 5

SHADED REGIONS Sometimes the GMAT asks you to find the area of a weird-looking region. The key to shaded-region questions is to focus on the non-shaded part. By subtracting the non-shaded portion from the total, you will arrive at the area of the shaded portion.

P

O

Q

R Weird shapes and figures are usually constructed by combining ordinary shapes in odd ways. Look for the shapes with which you’re familiar.

1. In the figure above, OPQR is a square. If O is the center of the circle, and the distance between point P and point R is 4 2 , what is the area of the shaded region? 16 - 8π 16 - 4π 32 - 8π 32 - 2π 64 - 12π

Shaded Area = Total Area - Unshaded Area

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DRILL

B S

10

y

A

R

x

z

T

1. In triangle RST in the figure above, does RS = RT? (1) y = 180 – 2x (2) x + y = y + z

C

3. In the figure above, ∠A = 90 and the length of chord BC = 10. Point A is the center of the circle. What is the area of the circle? 10π

2. A rectangular room is 12 feet long, 8 feet wide, x feet high, and has a volume of 1152 cubic feet. What is the greatest distance (in feet) between the center of the ceiling and any other point in the room?

10 3 π 25π 50π 100π

8 2 12

y

14 12 2 20

S

40 O

x

4. In the coordinate system above, the length of OS is 6. Which of the following must be true? I. The x-coordinate of point S is greater than -6. II. The slope of OS is greater than -1. III. The y-coordinate of point S is greater than -6. I only I and II only I and III only II and III only I, II, and III 376

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Class Lesson 5

R

Q

L

S

M

N

5. What is the circumference of the circle above? (1) The length of arc QRS is onesixth of the circumference. (2) The length of chord QS is 6. 6. The points (p, q) and (p - 2, q + c) are both on the line expressed by the equation y = 2x + 5. What is the value of c? -4 -1 1 2 4

B

E

C

8. In the figure above, L, M, and N are the centers of three circles, each with radius 4. What is the perimeter of the shaded region? π 4π 3 8π 3 4π 6π

A

D

7. In the parallelogram ABCD above, AD = 12. If the 3

the area of parallelo8 gram ABCD, then what is the length of EC? area of triangle ABE is

3 2 2 8 3 3 4

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ANALYSIS OF AN ARGUMENT Remember the basic approach discussed in the pre-class assignment: Step1:

Brainstorm

Step 2:

Outline

Step 3:

Write

Step 4:

Finish

Your task is to discuss how convincing you find the argument’s line of reasoning and the evidence supporting it. You also have the opportunity to suggest how the argument could be made convincing. You’ll receive a high score if you do the following: 1. Clearly identify the important features of the argument: conclusion, premises, and assumptions. 2. Critique the author’s logic and assumptions, but not the conclusion. Show that the assumptions could be wrong or that some key term is not adequately defined. 3. Suggest ways that the author could improve the argument. Explain the type of evidence that is needed to support the assumptions and fill in the gaps. 3. Do not give your opinion on the truth of the conclusion; that’s what the Analysis of an Issue essay is all about. Stick to the logic of the argument or you won’t get a good score. What does that really mean to you? Well, you finally have a chance to respond to their bogus arguments with your own thoughts and suggestions about what makes the argument bad. Now, let’s take a look at what the Argument essay is supposed to look like.

THE TEMPLATE All Argument essay topics are constructed in similar fashion. All Argument essays should follow a standard format as well. Let’s look at a good way to write an Analysis of an Argument essay.

Introduction Your introduction should quickly summarize the conclusion of the argument and present your position on the logical soundness of the argument. For example: The argument that (summary of argument) omits some important concerns that must be addressed to substantiate the argument. The author states that (restate argument). The conclusion is based upon assumptions that seem to be unsupported.

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Class Lesson 5

(Restate argument). This premise does not constitute a strong logical argument with reasonable support or proof for the main argument. It’s just that straightforward. This kind of writing may not look fancy, but at the same time, remember: You’re not trying to get published, you’re trying to express yourself clearly enough that you improve your chances of getting into the B-school of your choice, not someone else’s.

Body Paragraphs Now take the two–four strongest points you have brainstormed, and turn each into a paragraph. Make sure to use standard essay format and transition words. Each point you make should have about two sentences to support it. Assess the plausibility of the author’s assumption. Give some details or an example to support your assessment. You can also show how the author could improve the argument. Try using constructions such as these: The argument does not address the possibility that… The argument assumes that (assumption). It is possible, however, that…

Conclusion Write a strong conclusion paragraph. It should definitely begin with something like: In conclusion, … The above essay clearly demonstrates that… Summarize your critiques, and suggest ways for improvement. On this latter part, keep in mind that you need to have a certain “diplomatic tone” about your criticism. Be constructive, not destructive, in your criticism. Note how the essay in the pre-work points out not only what’s wrong with the author’s argument, but also how the author could improve the argument. It’s important that your essay come across this way as well if you want to improve your chances for a better score in the AWA.

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TEST SMARTS Adjust your approach to your place in the exam. 1. A circle is inscribed in triangle ABC such that point D lies on the circle and on line segment AC, point E lies on the circle and on line segment AB, and point F lies on the circle and on line segment BC. If line segment AB = 6, what is the area of the figure created by line segments AD, AE, and minor arc DE? 3 3−

9 4

π

3 3 −π 6 3 −π 9 3 − 3π It cannot be determined from the information given. Before you dive into a question, ask yourself: • Where am I on this test? • How should I approach this question? Questions 1–10

Questions 11–30

Questions 31–37

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PACING: THE MIDDLE Work the middle questions a little more quickly than the first ten questions, but not as quickly as those at the end of the exam. MATH Question numbers Score

1–10

11–20

21–30

31–37

Under 35

30 min.

25 min.

15 min.

5 min.

35–42

30 min.

20 min.

15 min.

10 min.

Above 42

25 min.

20 min.

20 min.

10 min.

VERBAL Question numbers Score

1–10

11–20

21–30

31–41

Under 28

30 min.

25 min.

10 min.

10 min.

28–34

27 min.

20 min.

18 min.

10 min.

Above 34

25 min.

20 min.

15 min.

15 min.

Your accuracy during the middle of a section still plays a large part in determining your final score. It will not affect your score as much as the first part of the test, but questions here have a bigger impact than do those in the last part of the test. Your pace should be similar to the pace at which you complete your homework. Move at a pace that allows you to read and work the questions carefully, but don’t spend too long on any one question. Most likely, you won’t need to pick up your pace on every question you do. Instead, avoid time sucks. Focus on the questions you know how to do, not the questions you don’t. Finally, think about your pacing for the entire group of questions. Focusing on an ideal time per question is a surefire way to increase anxiety, not raise your score. Some questions take more time to solve and some take less. That’s why we provide guidelines for how long sets of questions should take rather than time per question.

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TEST 3 GOALS Now that you’re becoming comfortable with the CAT format, work to increase your accuracy. Remedy any problems from your first CAT test. Here are a few goals: • Reduce the number of careless mistakes in the first 10 questions in each section. • Increase accuracy in the middle of the exam. Don’t get stuck on killer questions. Instead, focus on what you know. Try to do so within the suggested time frame. • Make sure to eliminate traps and ballpark rather than blindly guessing. • Based on your analysis of your last test, make any other necessary adjustments.

TEST ANALYSIS Pacing As you take the exam, look at the clock after you complete every 10 questions. Note the time remaining in the spaces below.

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Math

Verbal

#10______

#10______

#20______

#20______

#30______

#30______

Class Lesson 5

Accuracy After you complete the test, look at the score report. Count the number of questions right and wrong for the entire section. Then, count the number wrong for each portion of the exam.

T e s t A n a ly s is O v e r a ll S c o r e M a th S c o re V erb al S co re M a th S e c tio n P r o b le m s o lv in g % c o r r e c t D a ta s u ffic ic e n c y % c o r r e c t N u m b e r R ig h t in q u e s tio n s 1 – 1 0 N u m b e r R ig h t in q u e s tio n s 1 1 – 2 0 N u m b e r R ig h t in q u e s tio n s 2 1 – 3 0 N u m b e r R ig h t in q u e s tio n s 3 1 – 3 7 V e rb a l S e c tio n S e n te n c e C o r r e c tio n % c o r r e c t C r itic a l R e a s o n in g % c o r r e c t R e a d in g C o m p r e h e n s io n % c o r r e c t N u m b e r R ig h t in q u e s tio n s 1 – 1 0 N u m b e r R ig h t in q u e s tio n s 1 1 – 2 0 N u m b e r R ig h t in q u e s tio n s 2 1 – 3 0 N u m b e r R ig h t in q u e s tio n s 3 1 – 4 1

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Content Review the questions you missed and complete the log below. First, note the question number and question format (problem solving, data sufficiency, sentence correction, reading comprehension, or critical reasoning). Second, write down the question type/topic, the concepts or skills tested by the question. For example, a math question might test ratios, a sentence correction question might test idioms, or a critical reasoning question might test strengthening an argument. Finally, use the online explanations to determine why you missed the question. You might have made an error in reading the problem, performing calculations, or using a technique.

Question Number

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Question Format

Question Type

Diagnosis

Class Lesson 5

Action Plan What adjustments will you make to your pacing for the next test? Which techniques do you need to practice? Do you need to review any content? Do you have any questions for your instructor? Bring your score report to your next class. If there were any problems you couldn’t figure out after reviewing the explanations, print them out and bring them to class or extra help.

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HOMEWORK REVIEW Use this chart to note any questions you have from the reading or examples in the homework. Page #

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Question #

What question do you have?

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Class Lesson 5

PRACTICE B 105

1 30

A

1. What is the perimeter of the triangle above?

30

C

Note: Figure not drawn to scale.

3+ 3

3. In the triangle above, if BC = 4 2 , then what is the area of ∆ ABC? 64

2

3

2+ 2

16 + 16 3

3+ 3

8+8 3

3+ 5

8+4 2 8 2

P A (-3, 5)

2 2 R

Q

2 2

O

B

2. In ∆ PQR above, PQ = 1 2

2 2 1

2 2

4. In the figure above, AOB is the diameter of a circle centered at O. If the coordinates of A are (–3, 5), then the coordinates of B are (–3, –5) (–3, 5) (3, –5) (3, 5) (5, 3) 5. A circle with center (0, 0) and radius 8 will pass through all of the following points EXCEPT (–8, 0) (0, –8) (0, 8) (8, 0) (8, 8)

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6. The slope of a line containing points (2, –3) and (4, p) is –1. What is the value of p? –6 –5 1 5 12 B x

A

60

7. In the figure above, if A, B, and C are points on the circle, and if AB = AC, then what is the value of x? 40 45 50 55 60 N

M

8. In the figure above, if MN = 2, then the area of the circle with the center O is π 4 π 2

π 2π 4π

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r2 2

2r 2 r2 2r2 4r2 10. The length of an edge of cube A is 5% greater than the length of an edge of cube B. If the volume of cube B is 27 cubic centimeters, then what is nearest to the volume of cube A?

C

O

9. A square is inscribed in a circle with area πr2. What is the area of the square?

23.1 25.65 27.125 28.35 31.25

Class Lesson 5

11. For the line whose equation is

y + 2− b

= m+

2

,

x x m is not zero. If the line is rotated 90°, then the slope of that line would be 1 m 1

–

m

13. A certain cube floating in a bucket of water has between 80 and 85 percent of its volume below the surface of the water. If between 12 and 16 cubic centimeters of the cube’s volume is above the surface of the water, then the length of a side of the cube is approximately 4 5 7 8 9

m –m m–2

12. A sphere with a radius of 5 is hollowed out at the center. The part removed from the sphere has the

14. In the rectangle coordinate system, triangle ABC has a vertex at point (0, 56). If point B is at the origin, then how many points on line AC have integer values for both their x and y values? (1) The third vertex of triangle ABC lies on the x-axis, and the triangle has an area of 196. (2) Point A has a positive x-coordinate and a y-coordinate of zero.

same center, and a radius of 3. What fractional part of the original sphere remained? (The formula for the volume of a sphere is

2 5

V=

4 3

πr 3 .)

2 5 16 25 27 125 98 125 3 5

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ANSWERS AND EXPLANATIONS 1. D In the 30 : 60 : 90 triangle, the hypotenuse equals 2 and the long leg (the base) equals 2. C

9. D Draw the diagram, and make sure the square is

3.

inside the circle. Now, plug in r = 4. The area of the circle is 16π. The diameter of the circle is 8,

It’s a 45 : 45 : 90 triangle. The hypotenuse equals

which is also the diagonal of the square. Each

2 × 2 , or 1. 2

side measures 4 2 , so the area of the square is 32 (target answer).

3. C

When you drop a perpendicular from point B to side AC (call that point D), you create a 30:60:90 triangle (∆ABD) and a 45:45:90 triangle (∆BCD). Since BC = 4 2 , BD = AC = 4. Now look at the 30 : 60 : 90 triangle: Since BD = 4, AD = 4 3 . The base AC = 4 + 4 3 , and the height BD = 4. 1 Use the area formula: 4 + 4 3 (4) = 8 + 8 3 . 2

(

4. C

5. E

)

B is in the quadrant where x is positive and y is negative. The only choice with a positive x-coordinate and a negative y-coordinate is (C). Each of the points in the wrong answer choices is 8 units from the origin. However, the point (8, 8) is

8 2 units from the origin. (Draw the 45 : 45 : 90 triangle.) 6. B

Set up the slope formula, and solve: p − ( −3) p+3 = −1 , which becomes = −1, making 4−2 2 p = –5.

7. E

Since AB = AC, ∠B = ∠C. The measure of ∠A is 60°, so ∠B + ∠C = 120°. Thus, ∆ABC is equilateral, and each angle measures 60°.

8. D ∆ONM is a 45:45:90 triangle. Since MN = 2, you can calculate that ON = OM = 2. The area of the circle = π

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10. E

This is a very work-intensive problem, but there is no way around the work. First, find the side for cube B, which is 3 since its volume is 27. The length of the edge of cube A is 5 percent greater than the length of the edge of cube B. Thus, the length of the edge of cube A is 105 percent of cube B. We can find the length of the side of cube A by multiplying 3 times 1.05. To find the volume of cube A, we must raise the side of cube A to the power of 3. (3.15)3 = 31.25.

11. B Since we are given the equation of a line in a strange form, it’s probably a good idea to rewrite it in the more familiar form of y = mx + b. Rewriting the equation given reveals that we have been given the standard line equation. Now we know the slope is m, and the only question is what happens to it when it is rotated 90°. The best way to determine that might be to draw the picture of a line and then rotate it 90 degrees, which would reveal that the slope has become negative, which means we can eliminate choices (A) and (C). You can also eliminate choice (E) because for that choice to be correct the slope would not only rotate but would also change its value, and that does not happen here. The last two choices have only one difference: whether rotating the line makes the slope merely negative or the negative reciprocal. Plugging In should resolve that.

Class Lesson 5

12. D This one sounds weird, but don’t let yourself be

13. A

The best way to approach this problem is to Plug In the Answers since the answers give us the side of the cube. If we start with the middle choice, (C), then we have a cube with side 7. If the cube has a side of 7, then it will have a volume of 343. We are told that between 80% and 85% of the volume is below the surface of the water, which means that between 15% and 20% of the volume is above the surface. If the volume of the cube is 343, then 20% is about 68 and 15% is about 51. Neither of these numbers is between the 12 and 16 cubic centimeters that are supposed to be above water, so clearly this can’t be the answer. Since the numbers are too large, we need to try something smaller. Pick one of the smaller choices and try again.

14. A

Start by translating the question and understanding the pieces of the puzzle given and the pieces needed. The question tells us where two vertices are, point B at (0, 0) and another point at (0, 56). To answer the question, we need the coordinates of the last point. Statement (1) gives us the area of the triangle, which allows us to calculate the last side of the triangle, and also states that the triangle is a right triangle because the other leg lies on the x-axis. With this information we can find the slope of AC, which allows us to answer the question. Eliminate choices BCE, and keep choices AD. Statement (2) only tells us that the other leg of the triangle is on the x-axis but not how long it is, so it’s not sufficient.

thrown off by the inclusion of the sphere. Remember, any time ETS asks you about a strange figure, they have to supply you with the relevant formula, and here they do. From here, it’s a lot like a shaded region question. Start by figuring the area of the whole sphere, using the formula: It’s

500 π . Now, figure out the volume 3

of the smaller sphere. It’s 36π . Now, look carefully at the question: We need to determine what fractional part of the original sphere remains. First, determine the volume of the remaining portion of the sphere: 500 392 π − 36π = π. 3 3

To

determine

what

fractional part remains, we need to take the remaining portion and divide it by the original 392 π 392 98 volume: 3 . = = 500 500 125 π 3

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Unless your instructor has told you otherwise, you should take an online CAT before Class 6. Go to the Online Student Center at www.princetonreview.com. If you’re having trouble with any of the material you’ve covered so far, contact your instructor about an extra-help session. Feedback is a good thing. Call or e-mail us with any questions or comments you may have.

LESSON 6

SENTENCE CORRECTION REVISITED STYLE POINTS In addition to errors of grammar, stylistic mistakes sometimes appear in sentence correction questions. These style problems are not as important as the grammatical errors, so check for them only after you have eliminated all of the grammatical errors. The correct answer can contain a style mistake.

Short and Sweet If two answers both are grammatically correct, choose the shorter and simpler one. 1. The distribution of mass within the core of the Earth, like the mantle that surrounds the core, has been deduced from the orbital behavior of the Earth and the motions of satellites controlled by the Earth’s gravity. the mantle that surrounds the core that within the mantle surrounding the core that of the mantle surrounding the core the mantle the core surrounds the distribution of mass within the mantle that surrounds the core

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GMAT MANUAL

Passive vs. Active Voice If you have the choice, active voice is better than passive voice. 2. After Nixon spent months trying to counter mounting bad press and pressure from his own party, resignation was chosen by him instead of facing the impeachment process. resignation was chosen by him instead of facing the impeachment process resignation was chosen instead of impeachment resignation was the choice made by him rather than facing the impeachment process he chose to resign rather than face impeachment he chose resignation rather than being impeached

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Lesson 6

Redundancy Don’t be repetitive. Don’t be redundant. Don’t be repetitive. The reason for this is because you shouldn’t say the same thing more than once. 3. If the depletion of the ozone in the upper portion of the Earth’s atmosphere were to continue at its present rate, by the year 2000 the hole in the ozone layer would be at least one thousand miles wide or wider. If the depletion of the ozone in the upper portion of the Earth’s atmosphere were to continue at its present rate, by the year 2000 the hole in the ozone layer would be at least one thousand miles wide or wider. Were the depletion of ozone in the upper portion of the Earth’s atmosphere to continue at its present rate, by the year 2000 the hole in the ozone layer would be at least one thousand miles wide. Was the depletion of ozone in the upper portion of the Earth’s atmosphere to continue at its present rate, by the year 2000 the hole in the ozone layer would be at least one thousand miles wide or wider. If the depletion of ozone in the upper portion of the Earth’s atmosphere were continuing at its present rate, by the year 2000 the hole in the ozone layer would be at least one thousand miles wide. Should the depletion of ozone in the upper portion of the Earth’s atmosphere continue at its present rate, by the year 2000 the hole in the ozone layer would be at least one thousand miles wide or wider.

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GMAT MANUAL

SUBJUNCTIVE Use what you learned in the pre-class assignment to work the following questions. 1. The treaty specifies that the economy of the member nations have inflation rates of less than three percent and resist external tariffs. economy of the member nations have inflation rates of less than three percent and resist economy of each member nation have an inflation rate of less than three percent and be resistant to every economy of the member nations have an inflation rate of less than three percent and be resistant to economies of each member nation have inflation rates of three percent or less and are resistant to economies of every member nation have three percent or less of an inflation rate and resist 2. Implementation of the Worldwide Monetary Unit would begin as scheduled if the member nations will settle the dispute over each of the nations’ voting power in monetary policy issues. would begin as scheduled if the member nations will settle the dispute over each of the nations’ were to begin as scheduled if the member nations will settle the dispute about the nations’ would begin as scheduled if the member nations were to settle the dispute over each nation’s will begin as scheduled as the member nations settle the dispute about each nation’s would begin as scheduled when the member nations were settling the dispute over each nation’s

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Lesson 6

HARDER SENTENCE CORRECTION QUESTIONS The pre-class assignment discussed some common patterns that make questions difficult. Let’s use what you learned about those patterns to work some tricky sentence correction questions.

• If the sentence is hard to follow, look for specific errors and focus on only the relevant parts of the sentence. • If there are several good answers, look for subtle differences that make some answers worse than others. • If there appear to be no good answers, eliminate answers for breaking known rules, not because they sound bad. Use those concepts to work the following questions. 1. Since 1999, when Congress repealed the GlassSteagall Act, which prohibited commercial banks that engaged in investment banking, financial services companies began to spread their activities into all areas of banking, insurance, and securities operations. which prohibited commercial banks that engaged in investment banking, financial services companies began forbidding commercial banks from engaging in investment banking, financial services companies have begun which forbid commercial banks to engage in investment banking, financial services companies began that prohibited commercial banks from engaging in investment banking, financial services companies began which prohibited commercial banks from engaging in investment banking, financial services companies have begun

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GMAT MANUAL

2. The Gatherers, a lecture series which discusses life among early human tribes, suggests that hunting and foraging groups be viewed as distinct from agricultural communities, rather than as part of a continuum. which discusses life among early human tribes, suggests that hunting and foraging groups be viewed as distinct from agricultural communities, rather than as part of a continuum discussing life in early human tribes, suggests hunting and foraging groups should be viewed distinctly from agricultural communities, rather than continuously that discusses life in early human tribes, suggest that hunting and foraging groups are distinct from agricultural communities, rather than as part of a continuum discussing life in early human tribes, suggests that early hunting and foraging groups and agricultural communities be viewed as distinct, rather than as parts of a continuum discussing life among early human tribes, suggested to view hunting and gathering groups as distinct from agricultural communities, rather than in a continuum 3. Agricultural scientists have estimated that the annual loss by erosion of arable land caused by heavy rainfall and inadequate flood controls approaches two million acres per year. the annual loss by erosion of arable land caused by heavy rainfall and inadequate flood controls approaches two million acres per year the erosion of heavy rainfall and inadequate flood controls causes a loss of arable land approaching two million acres per year erosion caused by heavy rainfall and inadequate flood controls results in a loss of arable land approaching two million acres per year an annual loss approaching two million acres of arable land per year results from erosion caused by heavy rainfall and inadequate flood controls annually a loss of arable land approaching two million acres per year is caused by erosion due to heavy rainfall and inadequate flood controls

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Lesson 6

4. In the United States, less than $200 per capita is spent by the government each year to support arts and cultural institutions, such as the National Endowment for the Arts, although the amount in European countries is much greater. In the United States, less than $200 per capita is spent by the government each year to support arts and cultural institutions, such as the National Endowment for the Arts, although the amount in European countries is much greater. The United States government spends less than $200 per capita annually supporting arts and cultural institutions, like the National Endowment for the Arts, although European countries spend a much greater amount. Although European countries spend a much greater amount, in the United States, the government spends annually fewer than $200 per capita to support arts and cultural institutions, including the National Endowment for the Arts. Arts and cultural institutions, like the National Endowment for the Arts, are supported by the United States government spending less than $200 per capita each year, although the amount in European countries is much greater. Although European countries spend a greater amount, the United States government annually spends fewer than $200 per capita to support arts and cultural institutions, such as the National Endowment for the Arts.

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GMAT MANUAL

SENTENCE CORRECTION REVIEW To maximize your score on sentence correction questions, you must be extremely well versed in identifying common errors and 2/3 splits. If you’re running short on time or working on a killer question, take a few seconds to identify one error and eliminate based on it before you guess. Make sure you know how to identify the following common errors. You should also be familiar with the corresponding rules. Verb Tense ID: Rule: Parallel Construction ID: Rule: Idioms ID: Rule: Misplaced Modifers ID: Rule: Pronouns ID: Rule: Subject-Verb Agreement ID: Rule: Also, eliminate any words that have the word “being,” and if you are really in a crunch, choose the shortest answer.

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Lesson 6

MATH 6 FUNCTIONS As you learned in the pre-class assignment, function questions require you to follow the directions and plug the numbers into the equation. 1. The “superprime” of a number is defined as the sum of its distinct prime factors. What is the “superprime” of 40? 7 8 11 13 22 2. For every integer n, n* is defined as the sum of all the distinct factors of n. Which of the following is equal to 10*? 5* 15* 17* 18* 100*

SEQUENCES Sequence problems are often just repeating functions. You have to pull a number through the same function over and over. The directions can be confusing, and it’s tedious work, but keep track of your terms, and work through it.

Determine what place in the sequence you know and what place in the sequence you want to find.

3. A sequence of numbers satisfies the equation An = 2(An-1) + 1. If A4 = 10, what is the value of A1? 0.375 1.375 1.75 4.5 9

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GMAT MANUAL

Probability (at least one x) = 1 – Probability (no x at all)

GROUPS For group problems, decide whether to use the group equation or the group grid. 1. A group of fourth-graders and fifth-graders are going on a field trip to the zoo. Only 20% of the students remembered to bring written permission from their parents. If 40% of the students are fourth-graders and 25% of the fourth-graders remembered to bring written permission, what percentage of the students are fifth-graders who forgot to bring written permission? 10% 30% 50% 60% 80%

HARDER PROBABILITY QUESTIONS With “A or B” probabilities, don’t double-count things that are both A and B.

Previously, you learned a number of basic probability concepts: • All probabilities are between 0 and 1. • Probabilities are part-to-whole relationships that represent number of outcomes you want . number of total possible outcomes

• Probabilities can be expressed as fractions, decimals, or percents. • Probability (A and B) = Probability (A) × Probability (B) • Probability (A or B) = Probability (A) + Probability (B) 1. The integers from 1 to 100 inclusive are each written on a single slip of paper and dropped into a jar. If one slip of paper is removed at random, approximately what is the probability that the number on it is neither even nor a multiple of 3? 83% 67% 50% 33% 17% The probability that something does not happen is a useful concept. Remember that the probability that x happens and the probability that x doesn’t happen must add to 100%.

Probability (not x) = 1 – Probability (x)

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Lesson 6

2. The probability of rain on each of five days is

1

, except on the first

6 day, when it is

2

, and on the last day, when it is

5

4

. What is the

5

probability that rain will occur on at least one of the five days? 1 675 5 72 5 27 22 27 67 72 In some cases, drawing a probability tree can help you solve the problem. It shows all the possible outcomes. 3. Marco tosses a coin four times. What is the probability that he gets heads exactly twice? 1 4 3 8 7 16 1 2 5 8

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GMAT MANUAL

HARDER PERMUTATIONS AND COMBINATIONS Remember the permutation and combination formulas you learned earlier:

n! (n − r )! n! C= r!(n − r )! P=

n = number of candidates from which to choose r = number of items to be chosen

Tougher math problems mix permutations and combinations in a single problem or require you to adjust the formulas to account for the specifics of the problem. If some items are restricted, deal with them first.

1. Kyle is making a CD containing 10 of his favorite songs. He has chosen two blues songs and 8 rock-n-roll songs to be on the CD. The blues songs will be the first and last songs. How many different orderings of songs on the CD are possible? 3,628,800 161,280 80,640 20,160 160 2. Katie has 9 employees that she must assign to 3 different projects. If 3 employees are assigned to each project and no one is assigned to multiple projects, how many different combinations of project assignments are possible? 252 1,680 2,340 362,880 592,704 3. Coach Miller is filling out the starting lineup for his indoor soccer team. There are 10 boys on the team, and he must assign 6 starters to the following positions: 1 goalkeeper, 2 on defense, 2 in midfield, and 1 forward. Only 2 of the boys can play goalkeeper, and they cannot play any other positions. The other boys can each play any of the other positions. How many different groupings are possible? 60 210 2,580 3,360 151,200

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Lesson 6

4. Judges will select 5 finalists from the 7 contestants entered in a singing competition. The judges will then rank the contestants and award prizes to the 3 highest ranked contestants: a blue ribbon for first place, a red ribbon for second place, and a yellow ribbon for third place. How many different arrangements of prize-winners are possible? 10 21 210 420 1,260

ANALYSIS OF AN ISSUE Use the basic approach as your write your essay: Step 1:

Brainstorm

Step 2:

Outline

Step 3:

Write

Step 4:

Finish

In this essay, your task is to take a position on a topic. You will either agree or disagree with the statement made by the author. Use examples and detailed explanations to support your position. You’ll receive a high score if you do the following: 1. Pick a side. In real life, you may like to take the middle of the road and accept some of both sides, but that’s not the best way to approach an Issue essay. Pick one side and argue forcefully for it. 2. Back it up. Come up with two to four specific reasons why your position is correct. Make each one of these reasons a separate paragraph. 3. Get specific. Use concrete examples to support your points. Make them sound as relevant and factual as possible. Provide some detail. 4. Explain thoroughly. Don’t toss out a point and expect the reader to know exactly what you mean. Don’t assume any prior knowledge on the part of the reader. In fact, the grader will regard your paper more highly if he or she learns something from it. Now, let’s take a look at what the Issue essay is supposed to look like.

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THE TEMPLATE All Issue essay topics are constructed in a similar fashion. All Issue essays should follow a standard format as well. Let’s look at a good way to write an Analysis of an Issue essay.

Introduction Your introduction should quickly summarize the statement and present your position (for or against). For example: While many people feel that (summarize statement), the opposite is true. This essay will demonstrate why the latter position is correct. The author states that (summarize statement). This position is well justified. There are several reasons that this is true. It’s just that straightforward. This kind of writing may not look fancy, but at the same time, remember: You’re not trying to get published; you’re trying to express yourself clearly enough that you improve the chances of getting into the B-School of your choice, not someone else’s.

Body Paragraphs Now take the two to four strongest points you have brainstormed, and turn each into a paragraph. Make sure to use standard essay format and transition words. This is all about writing an essay the way you learned in the sixth grade. Each point you make should have about two sentences to support it. You’ll create a strong essay if you use specific, well-developed examples.

Conclusion Write a strong conclusion paragraph. It should definitely begin with something like: In conclusion, … The above essay clearly demonstrates that… Then, summarize your points and your position. That’s all there is to it.

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Lesson 6

TEST SMARTS You’ve developed a pacing plan for the beginning and middle parts of a section. Now it’s time to focus on the end of each section.

PACING: THE END MATH Question numbers Score

1–10

11–20

21–30

31–37

Under 35

30 min.

25 min.

15 min.

5 min.

35–42

30 min.

20 min.

15 min.

10 min.

Above 42

25 min.

20 min.

20 min.

10 min.

VERBAL Question numbers Score

1–10

11–20

21–30

31–41

Under 28

30 min.

25 min.

10 min.

10 min.

28–34

27 min.

20 min.

18 min.

10 min.

Above 34

25 min.

20 min.

15 min.

15 min.

Our pacing plan allots very little time to the final questions in each section. You may be wondering how you are going to do so much in so little time. Don’t worry. The final questions have the smallest effect on your score. The range of possible scores at this point in the exam is much smaller than the range of scores at the beginning of the exam. Though the final questions have a limited effect on your final score, they do have some effect. Maximize your final score by using the last few minutes wisely. Choose your battles. You probably won’t have time to attempt every question that you see. Don’t use up the remaining time on a time-intensive question. For example, you might see a must-be math problem with three statements. This is not be the wisest place to spend your remaining time. Eliminate what you can, guess, and move on to a question that takes less time to solve. Guess aggressively. Eliminate as many choices as possible before guessing. Every choice you eliminate increases your odds of getting a question right. Every additional question you get right helps your final score. Use what you’ve learned about recognizing trap answers.

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Answer every question. Do not let time run out without answering every question. You may not have time to work all the questions, but you must leave enough time to indicate an answer for every question. Leaving even one question unanswered can damage your score.

TEST 4 GOALS At this point, try to pull together all you have learned in this course. Your focus here should be not only on accuracy, but also on pacing. Put into practice all of the great strategies you have learned! Here are a few goals: • ELIMINATE careless mistakes in the beginning of the exam. • Reduce the number of careless mistakes in the middle of the exam. • Eliminate traps and ballpark at the end of the exam. • Don’t get stuck on killer questions. Remember, one in four questions doesn’t count anyway. • Keep track of your pacing; try to finish each chunk of questions within the recommended guidelines.

TEST ANALYSIS Pacing As you take the exam, look at the clock after you complete every 10 questions. Note the time remaining in the spaces below.

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Math

Verbal

#10______

#10______

#20______

#20______

#30______

#30______

Lesson 6

Accuracy After you complete the test, look at the score report. Count the number of questions right and wrong for the entire section. Then, count the number wrong for each portion of the exam.

T e s t A n a ly s is O v e r a ll S c o r e M a th S c o re V erb al S co re M a th S e c tio n P r o b le m s o lv in g % c o r r e c t D a ta s u ffic ic e n c y % c o r r e c t N u m b e r R ig h t in q u e s tio n s 1 – 1 0 N u m b e r R ig h t in q u e s tio n s 1 1 – 2 0 N u m b e r R ig h t in q u e s tio n s 2 1 – 3 0 N u m b e r R ig h t in q u e s tio n s 3 1 – 3 7 V e rb a l S e c tio n S e n te n c e C o r r e c tio n % c o r r e c t C r itic a l R e a s o n in g % c o r r e c t R e a d in g C o m p r e h e n s io n % c o r r e c t N u m b e r R ig h t in q u e s tio n s 1 – 1 0 N u m b e r R ig h t in q u e s tio n s 1 1 – 2 0 N u m b e r R ig h t in q u e s tio n s 2 1 – 3 0 N u m b e r R ig h t in q u e s tio n s 3 1 – 4 1

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Content Review the questions you missed and complete the log. First, note the question number and question format (problem solving, data sufficiency, sentence correction, reading comprehension, or critical reasoning). Second, write down the question type/topic, the concepts or skills tested by the question. For example, a math question might test ratios, a sentence correction question might test idioms, or a critical reasoning question might test strengthening an argument. Finally, use the online explanations to determine why you missed the question. You might have made an error in reading the problem, performing calculations, or using a technique.

Question Number

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Question Format

Question Type

Diagnosis

Lesson 6

Action Plan What final adjustments will you make to your pacing? Which techniques do you need to practice? Do you need to review any content? Do you have any questions for your instructor? Bring your score report to your next class. If there were any problems you couldn’t figure out after reviewing the explanations, print them out and bring them to class or extra help.

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HOMEWORK REVIEW Use this chart to note any questions you have from the reading or examples in the homework.

Page #

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Question #

What question do you have ?

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Lesson 6

PRACTICE 1. Of the 150 students at Hunter High, 45 are in the glee club and 72 are in the key club. If the number who are in neither group is twice the number who are in both groups, how many are in both groups? 22 33 44 55 66 2. If set N contains only consecutive positive integers, what is the sum of the numbers in set N? (1) Nineteen times the sum of the first number in the set and the last number in the set is 1729. (2) There are 38 numbers in the set. 3. If P is a set of consecutive integers, is there an even number of integers in set P? (1) The sum of the integers in set P is 0. (2) The product of the integers in set P is 0.

6. How many 4 digit numbers begin with a digit that is prime and end with a digit that is prime? 16 80 800 1440 1600 7. Chef Gundy is making a new “style” of salad which will contain two kinds of lettuce, one kind of tomato, one kind of pepper, and two kinds of squash. If Chef Gundy has 8 kinds of lettuce, 4 kinds of tomatoes, 5 types of peppers, and 4 kinds of squash from which to choose, then how many different “styles” of salad can he make? 640 1120 2240 3360 13440 8. Jean drew a gumball at random from a jar of pink

4. Alfred, ever hungry, decides to order 4 desserts after his meal. If there are 7 types of pie and 8 types of ice cream from which to choose, and Alfred will have at most two types of ice cream, how many distinct groups of desserts could he consume in his post-prandial frenzy? 588 868 903 1806 2010 5. Ten telegenic contestants with a variety of personality disorders are to be divided into two “tribes” of five members each, tribe A and tribe B, for a competition. How many distinct groupings of two tribes are possible? 120 126 252 1200 1260

and blue gumballs. Since the gumball she selected was blue and she wanted a pink one, she replaced it and drew another. The second gumball also happened to be blue and she replaced it as well. If the probability of her drawing the two blue 9 gumballs was , what is the probability that the 49 next one she draws will be pink? 1 49 4 7 3 7 16 49 40 49

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9. Kurt, a painter, has 9 jars of paint: 4 are yellow, 2 are red, and the remaining jars are brown. Kurt will combine 3 jars of paint into a new container to make a new color, which he will name according to the following conditions: Brun Y if the paint contains 2 jars of brown paint and no yellow. Brun X if the paint contains 3 jars of brown paint. Jaune X if the paint contains at least 2 jars of yellow. Jaune Y if the paint contains exactly 1 jar of yellow. What is the probability that the new color will be Jaune? 5 42 37 42 1 21 4 9 5 9

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10. When a die that has one of six consecutive integers on each of its sides is rolled twice, what is the probability of getting the number 1 on both rolls? (1) The probability of NOT getting an eight is 1. (2) The probability of NOT getting a seven is

25 36

.

Lesson 6

ANSWERS AND EXPLANATIONS 1. B Four words: Plug In the Answers. Of course, first you need to set up your group formula: Total = Group 1 + Group 2 – Both + Neither, so 150 = 45 + 72 – B + N. Start with C. If 44 students are in both groups, 88 would be in Neither. Does 150 = 45 + 72 – 44 + 88? No. That comes out to 150 = 161, clearly an incorrect formulation. You need a smaller number. Try 33, which would mean 66 would be in Neither group. Does 150 = 45 + 72 – 33 + 66? Absolutely. 2. C Statement (1) just lets you know that the sum of the first and last numbers in the set is 91, but that allows the first number to be 1 and the last to be 90 or the first to be 45 and the last to be 46, so it’s not sufficient. Statement (2) tells you how many numbers are in the set, but gives you no notion of the values of any of those numbers. Together, we know that there are 38 consecutive numbers, and that the sum of the smallest of the numbers and the largest is 91. This information is sufficient, since we now can determine exactly which two numbers are the smallest and largest in the set. 3. A Statement (1) lets us know that the sum of the consecutive integers in the set is 0. Since we’re dealing with consecutive integers here, we know some have to be positive and some have to be negative, and the positive and negative integers have to balance out (e.g., –1, 0, 1 would be a set that would work). The only way to have this balance of positive and negative integers is to have an odd number of integers, since 0 must also be included. Hence, Statement (1) is sufficient, and the answer must be (A) or (D). Statement (2) only lets us know that one of the numbers in the set is 0, but sheds no light on whether there is an odd or an even number of integers in the set. 4. C

5. C

Figure all the distinct combinations: PPPP (35), PPPI (280), and PPII (588). Add them together, and the total is 903.

6. E

This problem is a hidden combination problem. To find out how many different numbers will have a prime as the first digit and a prime as the last digit, we only to need find out how many different choices there are for each digit in the four digits. For the first and last digits, we have 4 different possible numbers (prime digits 2, 3, 5 and, 7). For the second and third digits, we have 10 possibilities (0–9, inclusive). If we multiply the possibilities for each digit (4 × 10 × 10 × 4), then we get the total number of combinations possible for the four-digit number.

7. D Just your average killer combination question. Find the number of ways you can choose 2 out of 8 lettuces, then 1 out of 4 tomatoes, then 1 out of 5 peppers, and lastly 2 out of 4 squash. Multiply them all together. The math would look  8 × 7  4 × 3 like this:   ( 4)( 5)  = 3, 360 .  2 × 1  2 × 1 8. B

It’s all about plugging in the answers here. Just be clear on what the answers represent: the chances of getting pink. Consider answer choice 3 C. If you have a chance of getting pink, that 7 4 means you have a chance of getting blue, 7 16 which would give you a chance of getting 49 two blue, which is too big. We need a smaller chance of getting blue, which actually means we need a larger chance of getting pink. The correct answer is (B).

You have to select five out of ten for tribe A, yielding 252 possible tribe A’s. Tribe B must consist of the remaining contestants, so there is just one possible tribe B.

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9. B

Don’t let the weird names confuse you. The question is really asking what the chances are of getting at least one yellow. That’s 1 minus the chances of getting no yellow. The probability of using three jars, none of which is yellow, is 5 Subtracting 5 from 1 leaves  5 4 3 .  × × = 42  9 8 7  42 37 , and the answer is (B). 42

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10. B

Taking a minute to translate and understand the pieces of the puzzle needed to answer the question reveals that to find out this probability, all we need to know is whether the number 1 is on the die. Statement (1) tells us that 8 is not on the die. The six integers could be either 0-5 or 27, so it’s insufficient. Eliminate AD. Statement (2) tells us that seven is on the die (since the probability is not 100% there is some chance of getting a 7) thus since all six integer are consecutive one cannot be on the die, thus statement 2 is sufficient.

Your course is wrapping up. Be sure to take another online CAT before Class 7. If you’re having trouble balancing your business school applications work with your business school applications work with your GMAT preparation, we can help. Got to www.princetonreview.com/ business for useful information. As always, let us know how things are going. Call or e-mail your local office with any questions or comments.

Lesson 7

LESSON 7

ARGUMENTS AND RC REVISITED CRITICAL REASONING REVIEW By now you should be a pro at identifying the different question types and the common wrong answer choices on critical reasoning questions. When running short on time or working on a killer question, be prepared to quickly identify the question type and eliminate the appropriate trap answers. Make sure you know the identification words and POE criteria for the types below. Weaken ID: POE: Strengthen ID: POE: Assumption ID: POE: Inference ID: POE:

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Resolve/Explain ID: POE: Identify the Reasoning ID: POE:

HARDER ARGUMENTS QUESTIONS Remember the factors that make verbal questions hard and the methods for coping with them. • If the argument is hard to follow, read carefully to find the important elements of the argument. • If there are several good answers, look for subtle reasons to eliminate answers, not reasons to keep them. • If there appear to be no good answers, double-check each answer for relevance to the conclusion. Use those concepts to work the following questions. 1. A recent study suggests that regular exercise improves the health of a person’s heart and cardiovascular system. Five years ago, people under the age of 60 accounted for 50 percent of the people who had suffered one or more heart attacks. Today, people under the age of 60 account for only 40 percent of the people who have suffered one or more heart attacks. The same study shows that people under the age of 60 exercise more regularly today than they did 5 years ago, while the exercise habits of people aged 60 and over have remained the same. Which one of the following most strengthens the argument? Some people over the age of 60 exercise as much or more than do people under the age of 60. The proportion of the population aged 60 and over has remained constant over the last five years. The use of cholesterol-lowering drugs has reduced the frequency of heart attacks among all age groups. People aged 60 and over are generally less capable of strenuous exercise than are people under the age of 60. A number of factors, such as nutrition and stress levels, affect the incidence of heart attacks.

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Lesson 7

2. Bands that are signed to recording contracts with major record labels generate higher average record sales than do bands who sign contracts with independent labels. The characteristics of the record label, such as marketing expertise and promotional budgets, must be a more important causative factor in record sales than are characteristics of the band, such as musical talent and ambition. Which of the following, if true, most seriously weakens the argument above? A band’s musical talent is a more important factor in selling records than its ambition. Some independent record labels have as much marketing expertise as do major labels. Consumers state that musical talent is their top criterion in determining which records to buy. Some bands signed to contracts with independent labels produce top-selling records. Major record labels base their decisions to sign bands on characteristics of the band, including musical talent and ambition. 3. Every driver in the United States is legally required to purchase liability insurance that protects other individuals in the event that the driver causes property damage or bodily injury. Some politicians argue that this insurance is partly responsible for the high rate of automobile collisions, because it reduces the drivers’ financial incentives to operate their automobiles in a safe and responsible manner. If drivers were required to pay directly for any damage they cause, they would drive more carefully. The politicians’ argument makes which of the following assumptions? If drivers were not required to carry liability insurance, individuals would be unprotected from uninsured drivers with little money to pay for damages they cause. Drivers who cause bodily injury to another feel little or no regret for their actions. Responsible drivers and reckless drivers pay similar premiums for liability insurance. The cost of liability insurance is more than some drivers can afford. Most drivers would purchase liability insurance even if they were not required to do so by law.

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4. In 1990, the number of new students admitted to Ph.D. programs each year averaged 1,250 students per university, while the total number of Ph.D. students enrolled averaged 7,500 students per university. By 2000, the number of new students admitted to Ph.D. programs each year had fallen to an average of 900 students per university, while the total number of Ph.D. students enrolled averaged 8,100 students per university. Which of the following conclusions is most strongly supported by the statements above? The total number of students enrolled in Ph.D. programs increased from 1990 to 2000. The average length of time a student remained enrolled in a Ph.D. program increased between 1990 and 2000. The percentage of applicants accepted by Ph.D. programs declined from 1990 to 2000. The number of universities remained constant from 1990 to 2000. The demand for Ph.D. degrees declined from 1990 to 2000.

HARDER READING COMPREHENSION Remember the factors that can make reading comprehension difficult and your strategies for attacking hard questions. • If the passage is hard to follow, identify the major elements, but don’t get lost in the details. • If there are several good answers, look for subtle reasons to eliminate answers, not reasons to keep them. • If there appear to be no good answers, read carefully to find answers that paraphrase information from the passage. Apply these concepts to the following passage and questions.

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Lesson 7

(5)

(10)

(15)

(20)

(25)

((30)

(35)

The tropical leaf-cutter, or attine, ant provides a remarkable example of complex symbiosis involving several species. The ants cultivate a fungus in underground caverns to serve as a source of food. DNA analysis reveals that the fungi in attine gardens around the world are clones of a single source. The ants do not allow their fungus crop to develop fruiting bodies, the means by which plants engage in sexual reproduction. Instead, a queen ant starting a new nest takes a sample of the fungus from the old nest to start the new garden, spreading the fungus vegetatively, or asexually. For many years, the phenomenon of sexual reproduction puzzled biologists, as it passes only half of the parents’ genes to the succeeding generation and requires a more complex mechanism than does asexual reproduction. What benefit of sexual reproduction would outweigh these limitations? One likely answer is that it provides a defense against parasitic attack. Simple parasites, such as bacteria or molds, mutate rapidly and pose a challenge to other species developing defense mechanisms. The rapid evolution of the attacking species provides an advantage in this biological arms race. Sexual reproduction allows the more complex species to shuffle its genes between generations and, as a result, evolve quickly enough to match the parasites’ mutations. A clonal monoculture, such as that of the attine, should be highly susceptible to parasites, yet preliminary research suggested no evidence of such a problem in the ants’ gardens. More comprehensive studies showed that the Escovopsis mold, a parasite related to the “green mold” known to commercial mushroom farmers, is present in the ants’ crops and poses a serious threat to the fungus. However, the attine ants provide the defense mechanism lacking in the fungus’s asexual reproduction by means of a bacterium that grows in patches on their skin. This actinomycete bacterium produces an antibiotic used to control the mold and limit its destructive effects on the ants’ food source.

1. The phrase “clonal monoculture” (line 26) refers to which of the following? The Escovopsis mold A species of leaf-cutter ant Sexual reproduction A fungus An antibiotic bacterium

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2. Which of the following can be most reasonably inferred from the passage? The fruiting bodies found on numerous fungi are a means of vegetative reproduction. Plants that rely on sexual reproduction are highly susceptible to parasitic attack by bacteria and molds. Sexual reproduction is an inefficient method for transmitting a parent’s genes to its offspring. The bacterium found on patches of green mold produces an antibiotic substance used by commercial mushroom farmers. Parasitic bacteria use sexual reproduction as a means to achieve rapid mutation. 3. The author describes the interaction between a parasite and a sexually reproducing organism as an “arms race” (line 22) in order to emphasize the aggressive nature of the parasitic organism warn of the dangers of biological weapons underscore the need for the attine ants to defend their gardens point out the devastating effects of the Escovopsis mold depict the result of rapid evolution by both organisms 4. The passage suggests which of the following about the fungus grown by the attine ants? By shuffling its genes, it is able to mutate rapidly in response to parasitic threats. Due to its asexual reproduction, it is susceptible to attack by the actinomycete bacterium. It could potentially engage in sexual reproduction. It is closely related to the crops grown by commercial mushroom farmers. Without the care of the attine ants, it would quickly become extinct.

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Lesson 7

MATH 7 In the pre-class assignment, you learned about some of the factors that can make math problems more difficult: • Complexity • Difficult phrasing • Tricks and Traps • Difficult topics Let’s take a closer look at how to use building blocks to solve complex problems.

COMPLEX PROBLEMS When dealing with complex math problems, simplify the task and break it down into manageable pieces.

• Use building blocks. Tackle the problem one step at a time. • Know your goal. Identify the missing piece needed to answer the question. B

A

D

C

1. In the figure above, ABC is an equilateral triangle with an area of 9 3 . B is a point on the circle with center D, and D is the midpoint of line segment AC. What is the area of the shaded region?

Geometry problems with multiple shapes are prime candidates for the building blocks approach.

18π − 9 3

9 3π 27π – 9 3 36π – 9 3 243π

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2. What is the product of the average (arithmetic mean) and the median of the set composed of the distinct prime factors of 4,095? 25 28 31 35 42 3. Al can complete a particular job in 8 hours. Boris can complete the same job in 5 hours. Cody can complete a second job, which requires twice as much work as the first, in 8 hours. If all three people work together on the larger job for 2 hours, how long, in hours, would it take Al, working alone, to finish the job? 0.8 3.0 6.8 8.0 8.8 4. If 66 × 103 × 152 = 2a × 3b × 5c, then which of the following is the value of 2

a × b 3 × c−2 ? –120 −

2 3

12 25 72 5 360

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Lesson 7

PRACTICE 1. This year at the Massachusetts Academy the boys are all 2, 3, 5, or 7 years of age. If the product of the ages of the boys in a given class is 10,500, then how many 5-year-olds are in that class? 0 3 5 125 2100 2. How many integers between 0 and 1570 have a prime tens digit and a prime units digit? 295 252 236 96 76 3. In March, Kurt ran an average of 1.5 miles an hour. If by June he had increased his pace by 10 seconds per mile, then which of the following expresses the number of hours it would take Kurt to complete one mile in June? 3590 602 2410

4. A teacher is assigning 6 students to one of three tasks. She will assign students in teams of at least one student, and all students will be assigned to teams. If each task will have exactly one team assigned to it, then which of the following are possible combinations of teams to tasks? I. 90 II. 60 III. 45 I only I and II only I and III only II and III only I, II, and III 5. On March 15th, the population of the city of Madrigoon was .15 billion people. On May 1st, an earthquake struck Madrigoon and destroyed .01% of the 30 million homes. If an equal number of people lived in each home and 50% of the people whose homes were destroyed moved to another city, then how many people moved to another city? 7.5 × 104 1.5 × 104 7.5 × 103 .15 × 104 .15 × 103

602 3890 602 3585 60 602 3590

6. If x is an integer greater than zero but less than integer n, is x is a factor of n? (1) n is divisible by all integers less than 10. (2) x is not a multiple of a prime number. 7. Dr. McCoy designed a space shuttle that can theoretically travel at a maximum velocity of 8 times the speed of light. If the speed of light is 300 million meters per second, then which of the following is the theoretical maximum speed, in meters per second, of Dr. McCoy’s shuttle? 2.4 × 103 2.4 × 108 2.4 × 109 3 × 106 3 × 109

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8. The product of all prime numbers less than 29 is approximately equal to which of the following? 2 × 104 2 × 106 2 × 108 2 × 109 2 × 1010 9. An ice cube is floating in a glass of water with between

1 6

and

1

of its mass above water and

7

the rest submerged below the water’s surface.

1

the part of the mass below water is between

45

and

5 1

1

1

6

40

1

2

7

9

5

5

and

6 5

and

6

7

6 and 7 6

and

7

7 6

10. Is (9x)3 – 2x= 1? (1) The product of x and positive integer y is not x. (2) x is a integer. 11. Of the 600 residents of Clermontville, 35 percent watch the television show Island Survival, 40 percent watch Lovelost Lawyers,and 50 percent watch Medical Emergency. If all residents watch at least one of these three shows and 18 percent watch exactly 2 of these shows, then how many Clermontville residents watch all of the shows? 150 108 42 21 –21

|

−1

 45 −1 + 5 −1  13.   a 10  

The ratio of the part of the mass above water to

1

430

12. A quarterly interest rate of 5 percent over a 12– month period is equal to an annual interest rate of approximately 60% 33% 22% 20% 15%

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45

(.009) 14. Which of the following is NOT equal to (.0003)

3

2.7 × 104 23 × 23 × 53 .00027 × 108 .033 × 108 1 3

−3

×

1 10 −3

3

?

Lesson 7

15.

3a−1 − 3 −1 a 3+a

17. Scott, Jean, and Warren are all building wooden

=

models for an architectural presentation at noon 3−a

tomorrow. If their individual probabilities of 1 1 finishing on time are x, , and , respectively, 3 7 then what is the probability that Warren will finish

3a 3+a 3a 3a−2 + 1

3a 3−a

on time but Jean and Scott will not? 21x − 38

a

21

a

12 x

1+ a

21 2 − 2x

16. On this year’s Westchester basketball team, the players are all either 5, 7, or 11 years of age. If the product of the ages of the players on the team is 18,865, then what is the probability that a randomly selected team member will NOT be 7?

21 2x − 2 21

3

x

7

21

2 5 16 37 3 5 49 55

18. If x is greater than 0 but less than 10 and k = x9, what is the value of integer k? (1) x2 has a units digit of 1. (2) x –2 <

1 50

19. If a is a positive integer, is a + b an even integer? (1) xaxb = 1 (2) x ≠ 1 20. If a = (23)(43)(59) and b = (46)(56)(69), then which of 3

the following values is less than ab ? (27)(55)(63) (2)(43)(55)(63) (210)(33)(55) (212)(55)(6) (26)(53)(67) 21. Is the tens digit of two-digit positive integer p divisible by 3? (1) p – 5 is a multiple of 3. (2) p – 11 is a multiple of 3.

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22. Entries in a particular lottery game are made up of three digits, each 0 through 9. If the order of digits in the entries matters, how many different possible entries exist in which all three digits are not equal? 516 720 989 990 1321

23. A baseball team consists of 20 players, 5 of whom are pitchers and 15 of whom are position players. If the batting order consists of 8 different position players and 1 pitcher, and if the pitcher always bats last in the order, then which of the following expressions gives the number of possible different batting orders for this baseball team?

(15!)(5) 8!

(15!)(5) 7!

(15!)(5!) 7!

(15!)(5) 20!

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Lesson 7

ANSWERS AND EXPLANATIONS 1. B

The phrase “product of the ages . . . is 10,500” tells us that we need to factor 10,500 to see what its prime factors are. The fact that all the ages given are prime numbers tells us that we want to use a factor tree. If we break 10,500 down to its prime factorization, we get 2 × 2 × 3 × 5 × 5 × 7. There are three five-year-olds.

2. B

First, there are only 4 prime digits: 2, 3, 5, and 7. Next, if you start writing down the numbers that meet the question’s criteria, you will see a pattern emerge. Between 0 and 99, the only numbers that will work are: 22, 23, 25, 27, 32, 33, 35, 37, 52, 53, 55, 57, 72, 73, 75, and 77, for a total of 16 numbers. Between 100 and 199, the only numbers that will work are: 122, 123, 125, 127, 132, 133, 135, 137, 152, 153, 155, 157, 172, 173, 175, and 177; a total of 16 numbers. The pattern becomes clear; in every hundred, there are 16 numbers that we want. Since there are 15 hundreds between 0 and 1500, so far we have 15 × 16 = 240 numbers. Lastly, we need to count the numbers from 1501 to 1570 that meet the question’s requirement. Those are: 1522, 1523, 1525, 1527, 1532, 1533, 1535, 1537, 1552, 1553, 1555, and 1557 for an additional 12 numbers. 240 + 12 = 252.

3. C

The problem has two conversions to watch out for; first, it gives 1.5 miles in March but 1 mile in June second, it adds 10 seconds to his mile per hour rate. The order in which you deal with these are up to you, but they must be dealt with. First let’s deal with the 1.5 mile to 1 mile problem. Initially, he runs 1.5 miles per hour, which is the same as saying that he does 3 halves of a mile in 60 minutes, thus each half must take 20 minutes. Now we know that in March it took him 40 minutes to run a mile. Let’s now convert those minutes to seconds, 40 minutes = 2400 seconds. If by June he increased his pace by 10 seconds, that means it would take him less time to complete the mile, so in June a mile would take him 2390 seconds. Now we have the time it would take him to do a mile in June, so the last step is to convert 2390 seconds to hours. To do so we must divide 2390 by 60 to get minutes and then divide it again by 60 to convert minutes into hours.

4. B

This is a combination problem. The one wicked twist in the problem is that they have not told you how many members are on each team, thus allowing you to get several different answers. The best way to approach this problem is to try out the different possible ways of arranging the team members: You could have teams with equal numbers (2 on a team), you could have 3-, 2- and 1-member teams, or you could have 4, 1 and 1 member teams. Now just figure out the possibilities out for each of these options, and eliminate answers appropriately.

5. C

You need to start by figuring out how many people are in each home; we have 150 million people in 30 million houses. Just ignore all those extra zeros, and you’ll realize you need to divide 15 by 3, which means there are 5 people per home. Next, we need to figure out how many homes were destroyed; .01 percent of 30 million is 3,000. Now, half of the inhabitants of the destroyed homes decided to move away; if there are 5 people per home, then there were 15,000 people in the destroyed homes. Half of them left the city, so 7,500 left. Now translate into scientific notation: 7,500 = 7.5 × 103.

6. B

Statement (1) tells us that n is a multiple of all numbers 1–9, inclusive. This does not tell us if n is also a multiple of 11 (or any other prime number greater than 9). x could be 5 or it could be 11, so we don’t know whether it’s a factor of n. Statement (1) is insufficient. Eliminate AD. Statement (2) tells us that x is not a multiple of a prime number, but all integers greater than 0 are multiples of prime numbers except for 1, so what Statement (2) really tells us is that x is 1. And 1 is a factor of all integers, so x must be a factor of n.

7. C

Given the fact that the answers here are in scientific notation, we ought to do our calculations in scientific notation as well. 300 million is 3.0 × 108. Multiply that by 8 and you get 2.4 × 108 or 24 × 109.

8. C

First, write out all primes less than 29 (2, 3, 5, 7, 11, 13, 17, 19, 23), then begin grouping them as numbers whose products are multiples of 10 (because the answers are expressed as powers of 10). You get (2 × 5)(3 × 7)(11)(13)(17)(19)(23) which could be expressed as 10 × 21 × 11 × 13 ×

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17 × 19 × 23. Then round these products as close to multiples of 10 as possible, which gives us 10 × 20 × 10 × 10 × 20 × 20 × 20. Each 20 can be expressed as 2 × 10, yielding 10 × 2 × 10 × 10 × 10 × 2 × 10 × 2 × 10 × 2 × 10. Expressing the products as exponents yields 24 × 107. Finally, 24 = 16, which is approximately 20 or 2 × 10. So we have 2 × 10 × 107, or 2 × 108. 9. A

First, express the ratio of mass above to the mass

below when

12. C

Just ballpark questions like these. Compound interest is always a little more than simple interest. Simple interest at 5 percent per quarter would be 20 percent. Compound interest would be a little more than 20 percent. The only possible answer is (C).

13. E

This question tests basic math in a somewhat

1 of the mass is above the water: 6

1 6 = 1 . Now, repeat the process for when 1 of 5 5 7 6 1 1 7 the mass is above the water: = . 6 6 7

10. C

subtract them twice, otherwise we’re doublecounting them. From here, it’s just a question of getting your numbers. The total is 600. Group 1 is 210, group 2 is 240, and group 3 is 300. There is no “Neither” (everyone watches at least one show), and 108 people watch exactly two shows. Now just plug in: 600 = 210 + 240 + 300 – 108 – 2x + 0. Now solve for x, and you have the number of people that watch all three shows.

Start by translating the question and understanding the pieces of the puzzle given and

complex manner, combining exponent and fraction rules. First, we should probably reexpress the numbers with negative exponents as

the pieces needed. A little working of the question reveals that the only way to make the equation equal 1 is for 9 to be raised to the power of 0. For 3 that to happen, x must either be 0 or . Statement 2 (1) tells us that x is not 0, but it doesn’t tell us 3 whether x is an integer or whether it could be . 2 Eliminate AD. Statement (2) says that x is an 3 integer, so it can’t be but it could still be 0 or 2 some other integer; thus this statement alone is not sufficient. Eliminate choice (B). Together, we know that x is neither an integer nor 0, so there’s

fractions: 45 −1 =

1 1 and 5 −1 = . Add the 45 5

fractions together, and you get

10 2 or . Next, 45 9

remember that dividing by 10 is the same as multiplying by

1 2 1 1 . × = . Now, we have 10 9 10 45

to deal with the final negative exponent:

 1    45 

−1

=

1 = 45 . 1 45

no way that the equation can equal 1. 11. D Remember the group formula? Total = Group 1 + Group 2 – Both + Neither? Well, this is just a ramped up group problem. Here, we have three groups instead of two. With three groups, the formula undergoes a slight modification: Total = Group 1 + Group 2 + Group 3 – (number that is in two groups) – (2)(number that is in all three groups) + Neither. Ugh. It makes sense though: If someone is in all three groups, we have to

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14. D The answers here indicate that you need to express the fraction given as scientific notation or you can try to multiply out the entire fraction, which would be much more difficult. First, let’s work with the numbers inside the parentheses,

Lesson 7

.009 can be expressed as 9 × 10–3 and .0003 can be

don’t need to do anything but use that number.

expressed as 3 × 10–4. Next, raise the numbers

1 , then 3 2 the probability of her not finishing would be . 3 1 Plug in a value for Scott: Let x = . So his chances 4 3 of not finishing would be . Now just multiply 4 1 2 3 1 the whole mess together: × × = . Now 7 3 4 14 1 just plug in for x, and find the answer that 4 For Jean, if her probability of finishing is

inside the parentheses to the exponents outside 9 3 × 10 −9 the parentheses; we now have 3 . You 3 × 10 −12 need to re-express 93 in terms of 3: 93 = 36. Now divide, remembering that you should subtract exponents when you divide. This yields 33 × 103, or 27,000. Now find the answer that is not 33 × 103, or 27,000. 15. A

Plug in. Let a = 2, then solve the given equation. 1 3 2 5  1 3 3a–1 = 3  = . 3–1a = × 2. − = , which,  2 2 3 2 3 6 1 when divided by 5 (the sum of 3 + a) is . Now 6

matches. 18. C

For Statement (1), simply square the numbers between 0 and 10; the results show that x could be either 1 or 9, thus there are at least 2 possible values for k. Eliminate AD. The only values of x that work with Statement (2) are 8 and 9, so Statement (2) is not sufficient, but the two statements together make it clear that the value of x is 9.

19. C

Start by translating the question and understanding the pieces of the puzzle given and the pieces needed. To answer this question, we need to know whether b is a positive integer or not. Statement (1) doesn’t give enough information to figure out whether x = 1 or whether a + b = 0. Statement (1) is thus insufficient; eliminate AD. Statement (2) alone tells nothing about a or b, so it’s not sufficient. Eliminate choice (B), and keep CE. Considered together, we know that x is not 1, and thus a + b must equal 0. Since xaxb = xa + b = 1, we know that a + b must be equal to 0.

20. C

This question is all about simplification. First, translate both a and b into their simplest forms: a = (2 3 )(4 3 )(5 9 ) = (2 3 )(2 6 )(5 9 ) = (2 9 )(5 9 ) and b = (46)(56)(69) = (212)(56)(29 × 39) = (221)(56)(39). So ab = (230)(515)(39) and 3 ab = (210)(55)(33). Yeesh. Now just compare that number with your answer choices. Remember that 6 is the same as 2 × 3, so 63, for example, is the same as (23 × 33).

just plug 2 into each answer choice, and find which one works. 16. B

17. C

Before the probability can be found, you need to know how many 7-year-olds are on the team and how many total members the team has. The phrase “the product of the ages of the players . . .” gives us the hint that we will need to factor 18,865 to find the age distribution of the team members. Also, since the ages given are all prime numbers, you should realize that a factor tree will help a lot here. The prime factorization of 18,865 is 5 × 7 × 7 × 7 × 11, so there must be five children on the team whose ages are 5, 7, 7, 7, and 11. So the probability of selecting a child that is not age 7 will be two (because two of the children are not age 7) out of five (because there are 5 total children from whom to choose ). To find the probability here, we just need to multiply the probability that Warren finishes by the probability that Scott doesn’t finish by the probability that Jean doesn’t finish. Since they have given us the probabilities for Warren already, we

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21. E

We need to plug in here. For Statement (1), p could be 38, which would produce a “yes”, or 47, which would produce a “no”. For Statement (2), p could still be 38, which would produce a “yes”, or 47, which would produce a “no”. Since we were able to use the same numbers in each statement, we know that they aren’t sufficient together, either. The correct answer is (E).

22. D For each entry, order matters and we have a choice from among 10 possibilities for each digit, so the total number of possible entries is 103 = 1000. For each integer, there is exactly one entry for which all the digits are the same. So the total number of entries with the digits not all the same is 1000 – 10 = 990.

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23. B

We are choosing 8 of the 15 position players, and order matters. So the number of possible orders is =

15! 15! = . We are choosing any 1 of the (15 − 8)! 7!

5 pitchers, so we multiply the result we got in the prior step by 5 to get our answer.

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